text:
\section{Introduction}
In this paper we denote the $2$-dimensional annulus by ${\mathbb A} = {\mathbb R} / {\mathbb Z} \times [0,1]$ equipped with the standard area-form $\omega$.
We let $F^\infty_{{\mathbb A}}$ denote the set of $\omega$-preserving $C^\infty$ {\it pseudo-rotations} of ${\mathbb A}$ (the precise definition will appear in Definition \ref{defi prot}). Namely, we set
\begin{eqnarray*}
F^{\infty}_{{\mathbb A}} := \{ f \in \diff^\infty({\mathbb A}, \omega) \mid \mbox{$f$ is isotopic to {\rm Id} and has no periodic points} \}.
\end{eqnarray*}
The study of pseudo-rotations can be essentially traced back to the question of Birkhoff \cite{Bir} (see also \cite{H98}) as to whether there are non-trivial analytic diffeomorphisms of the $2$-sphere with $2$ fixed points (the existence of such diffeomorphisms was recently announced by Berger \cite{Ber}).
The name \lq\lq pseudo-rotation\rq\rq was introduced by B\'eguin, Crovisier, Le Roux and Patou in \cite{BCLP}.
By a result of Franks \cite{F88},  for $\omega$-preserving homeomorphisms of ${\mathbb A}$, the notion of irrational pseudo-rotations in \cite{BCLP} coincides with ours.
In particular, each  $f \in F^\infty_{{\mathbb A}}$ admits a {\it rotation number} $\rho(f) \in (0,1)/{\mathbb Q}$ (see Definition \ref{def rotation number} for the details).
For each ${\alpha}\in(0,1) \setminus {\mathbb Q}$ we set
\begin{eqnarray*}
F^{\infty}_{{\mathbb A}}({\alpha}) &:=& \{ f \in F^\infty_{\mathbb A} \mid \rho(f) = {\alpha} \}, \\
O^{\infty}_{{\mathbb A}}({\alpha}) &:=& \{ hR_{\alpha} h^{-1} \mid h  \in \diff^\infty({\mathbb A}, \omega) \}
\end{eqnarray*}
where $R_{\alpha}$ denotes the rotation $(x,y)\mapsto(x+{\alpha},y)$.
It is clear that $O^{\infty}_{{\mathbb A}}({\alpha}) \subset F^{\infty}_{{\mathbb A}}({\alpha})$ for any ${\alpha} \in (0,1) \setminus {\mathbb Q}$.  It is not hard to see that $F^{\infty}_{{\mathbb A}}({\alpha})$ is closed in the $C^\infty$-topology for each irrational ${\alpha}$.    In particular, $\overline{O^{\infty}_{{\mathbb A}}({\alpha})}\subset F^{\infty}_{{\mathbb A}}({\alpha})$ for any ${\alpha} \in (0,1) \setminus {\mathbb Q}$, where the closure is taken in the $C^\infty$-topology.
Our main result is the following.

\begin{thm}\label{main thm F = closure O}
	For a Baire generic ${\alpha} \in (0,1) \setminus {\mathbb Q}$, we have
	\begin{eqnarray*}
	F^{\infty}_{{\mathbb A}}({\alpha}) = \overline{O^{\infty}_{{\mathbb A}}({\alpha})}
	\end{eqnarray*}
	where the closure is taken in $\diff^\infty({\mathbb A})$.
\end{thm}

In other words, for a Baire generic ${\alpha}$, any pseudo-rotation $f$ with rotation number ${\alpha}$ is the $C^\infty$-limit of a sequence $f_k$ of area preserving diffeomorphisms, which up to a smooth area preserving change of coordinates, is the standard rotation $R_{\alpha}$.
In particular $f$ is approximable by {\it integrable systems}.

We can see Theorem \ref{main thm F = closure O} as a natural analogue of a well-known theorem of Herman in \cite{H}. Recall that one of the most prominent results in the study of circle diffeomorphisms is the following.

\begin{thm}[Herman-Yoccoz]\label{thm HY}
	For any irrational ${\alpha} \in {\mathbb R}/{\mathbb Z}$, we denote by $F^\infty({\alpha})$ the set of $C^\infty$ circle diffeomorphisms with rotation number ${\alpha}$, and denote by $O^\infty({\alpha})$ the set of $C^\infty$ circle diffeomorphisms which are $C^\infty$-conjugate to the standard rotation $R_{\alpha}$. Then we have
	\begin{eqnarray*}
	F^\infty({\alpha}) = \begin{cases}
		\overline{O^\infty({\alpha})} =  O^\infty({\alpha}) & \mbox{if ${\alpha}$ is Diophantine}, \\
		 \overline{O^\infty({\alpha})} \neq O^\infty({\alpha})  & \mbox{if ${\alpha}$ is Liouville}.
	\end{cases}
	\end{eqnarray*}
	Here in the above the closures are taken under the $C^\infty$-topology.
\end{thm}

The above result for Diophantine ${\alpha}$ was conjectured by Arnold, who showed in \cite{Arn} that any $C^\omega$ circle diffeomorphism with a Diophantine rotation number ${\alpha}$ which is sufficiently close to $R_{\alpha}$ in the $C^\omega$-topology, is infact $C^\omega$-conjugate to $R_{\alpha}$. Arnold's result was then generalised to the $C^\infty$-category by Moser in \cite{Mos}.
This is the beginning of what is now known as the Kolmogorov-Arnold-Moser theory. The global picture was for the first time established by Herman in the seminal paper \cite{H}. In \cite{H}, Theorem \ref{thm HY} was proved for a subset of ${\alpha}$ with full Lebesgue measure. Khanin and Sinai gave in \cite{KS} a different proof  of the main result in \cite{H}  building on a renormalization theory for circle diffeomorphisms. The Diophantine part of Theorem \ref{thm HY} was completed by Yoccoz in \cite{Yo84}. We also mention Katznelson-Ornstein's papers \cite{KO1,KO2} on circle diffeomorphisms with low regularity, and Yoccoz's paper in \cite{EKMY} on $C^\omega$-linearization under the sharp arithmetic condition, i.e., ${\cal H}$-condition. For a recent survey of this development and beyond, we refer the reader to \cite{EFK}.

The Liouville part of Theorem \ref{thm HY} was conjectured by Herman in \cite[Conjecture 7.1]{H}. In fact Herman already showed that Theorem \ref{thm HY} holds for a Baire generic set of ${\alpha}$, see \cite[Theorem 7.3]{H}. However, his proof was based on the Diophantine part of Theorem \ref{thm HY} (at least for a full measure set of ${\alpha}$), and used certain properties of the function $t \mapsto \rho(R_t f)$ of a circle diffeomorphism $f$. It is still an open question of Herman whether the Diophantine rigidity holds within pseudo-rotations. Moreover, it is unclear how to deform a general pseudo-rotation within the set of pseudo-rotations, and change the rotation number. This blocks a direct generalisation of Herman's approach for pseudo-rotations. The full answer to Herman's conjecture was provided by Yoccoz in \cite{Yo95}. Yoccoz showed that any $C^\infty$ circle diffeomorphsim with a Liouville rotation number can be $C^\infty$-approximated by a {\it quasi-rotation}: this is a class of circle diffeomorphisms which, among other things, admits a renormalization that {\it is} a standard rotation.
Our proof of Theorem \ref{main thm F = closure O} is somewhat similar to the proof of Yoccoz: we also consider certain renormalizations of a pseudo-rotation. However, the type of estimates are very different. We are unable to transfer the strong estimates for circle diffeomorphisms, such as Denjoy's inequality in \cite{Yo84}, to general pseudo-rotations, due to the possible occurrence of complicated geometry which does not appear in dimension $1$. On the other hand, the area-preserving hypothesis provides us with certain strong $C^0$-estimates established in \cite{A}. Combining such estimates with a suitable arithmetic condition, we are able to extract some useful information from a sequence of suitably renormalized pseudo-rotations.

We can also compare Theorem \ref{main thm F = closure O} with the main result in \cite{B2}, which says that any smooth area preserving pseudo-rotation $f$ on the closed $2$-disc, meaning that $f$ fixes the origin and has no periodic points on the annulus complementary to the origin, is the $C^0$-limit of smooth periodic disc maps $f_k$, that each fix the origin.
 In the latter there are no restrictions on the (irrational) rotation number of $f$, but in this current paper our integrable approximations are in
every way stronger: 1) In \cite{B2} the sequence of approximations $f_k$, while $C^\infty$-smooth,
only converge in the $C^0$ topology to $f$.  2) In \cite{B2} the $f_k$'s are not necessarily area preserving.  3)  In \cite{B2} the approximation maps $f_k$ have rational rotation numbers $p_k/q_k\in{\mathbb Q}$ converging to the rotation number $\alpha$ of $f$ as $k\rightarrow\infty$.
One cannot perturb the $f_k$ in \cite{B2} in an obvious way to make the rotation number equal to $\alpha$ while retaining closeness to $f$.   On the other hand, it is easy to modify the $f_k$'s in the current article, if one so wishes, to make the
rotation number rational and keep the closeness to $f$.   In short, the fact that in this paper we are able to find integrable approximations without altering the rotation number is also a stronger conclusion than in \cite{B2}.

Another motivation behind our result is the important work of Anosov and Katok \cite{AK70} and the extensions in Fayad-Saprykina \cite{FS}, see also Fayad-Katok \cite{FK}, in which, for generic rotation numbers, more precisely all Liouville rotation numbers,
examples of pseudo-rotations are constructed which are dynamically interesting, that is, not conjugate to a rotation.    These \lq\lq exotic\rq\rq pseudo-rotations of Anosov-Katok lie, by construction, in the $C^\infty$-closure of $\cup_{t\in{\mathbb Q}}O^{\infty}_{{\mathbb A}}(t)$
rather than the closure of $O^{\infty}_{{\mathbb A}}({\alpha})$.   It however follows from our main result that, for a possibly smaller Baire-generic set of rotation numbers than the Liouville numbers, that the Anosov-Katok constructions do indeed lie in the closure of
$O^{\infty}_{{\mathbb A}}({\alpha})$ with \textit{fixed} rotation number ${\alpha}$. Moreover, we have the following interesting corollary.

\begin{cor}\label{cor wm}
For a Baire generic ${\alpha} \in (0,1) \setminus {\mathbb Q}$, 	the set of weakly mixing pseudo-rotations in $F^{\infty}_{{\mathbb A}}({\alpha})$ forms a Baire set with empty interior, with respect to the $C^\infty$ topology.
\end{cor}
\begin{proof}
	On the one hand, Anosov and Katok, see \cite{AK70}, show that for a Baire generic ${\alpha}$, weak mixing is a $C^\infty$-generic property in $\overline{O^{\infty}_{{\mathbb A}}({\alpha})}$.   Thus by Theorem \ref{main thm F = closure O}
	weak mixing is a $C^\infty$-generic property in $F^{\infty}_{{\mathbb A}}({\alpha})$.

	On the other hand, the second statement follows since elements of $O^{\infty}_{{\mathbb A}}({\alpha})$ are never weak mixing and by Theorem \ref{main thm F = closure O} the complement $F^{\infty}_{{\mathbb A}}({\alpha})\backslash O^{\infty}_{{\mathbb A}}({\alpha})$
	has empty interior.
\end{proof}
\begin{rema}
	Corollary \ref{cor wm} is seen to be rather sharp in the following sense:
	\begin{enumerate}
	\item the genericity of ${\alpha}$  cannot be improved into any subset of $(0,1)$ with positive Lebesgue measure. This follows from the KAM result of Fayad-Krikorian \cite{FKrik} (attributed by the authors to Herman), that a neighborhood of $R_{\alpha}$ in 	$F^\infty_{{\mathbb A}}({\alpha})$ lies in $O^\infty_{{\mathbb A}}({\alpha})$, for any Diophantine ${\alpha}$.
	\item Weakly mixing cannot be replaced by mixing. In fact, it follows from the proof of \cite{Br} and \cite{A} that for a Baire generic ${\alpha}$, $F^\infty_{\mathbb A}({\alpha})$ contains no topologically mixing maps. See also Theorem \ref{thm aflxz}.
	\end{enumerate}
\end{rema}

Recently, Avila and Krikorian have announced \footnote{See the minicourse of Krikorian in the program \lq\lq Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory \rq\rq in 2020 at Simons Center for Geometry and Physics.} an improvement of Theorem \ref{main thm F = closure O}: for every non-Brjuno $\alpha$, one has  $F^{\infty}_{{\mathbb A}}({\alpha}) = \overline{O^{\infty}_{{\mathbb A}}({\alpha})}$. Moreover, they have announced the following result:  for every pseudo-rotation $f$ in an open neighborhood of the rigid rotations on ${\mathbb D}$, there exists a sequence of area-preserving diffeomorphism $h_n$ such that $h_n f h_n^{-1}$ converges to $R_{\rho(f)}$ in the $C^\infty$ topology.
Their method involves delicate estimates on high iterates of the maps, while our method for getting this weaker result relies only on rather soft estimates.

\subsection*{Acknowledgements}

Z.Z. would like to thank Artur Avila and Rapha\"el Krikorian for discussion on one occasion.
Z.Z. would also like to acknowledge the online talk by Rapha\"el Krikorian during the Workshop \lq\lq Between Dynamics and Spectral Theory \rq\rq at  the
Simons Center for Geometry and Physics back in 2016, which inspired this article.
This work was initiated in 2019 while the authors were at the Institute for Advanced Study both supported by the National Science Foundation under Grant No.\ DMS-1638352.
We thank them for their hospitality and excellent working environment.
B.B. was also partially supported by the SFB/TRR 191 `Symplectic Structures in Geometry, Algebra and Dynamics', funded by the DFG (B1 281071066 -- TRR 191)

\section*{Notation}

In the rest of this paper, we use the following notation.
For a subset $A \subset {\mathbb R}^2$ we denote by $Int(A)$ the interior of $A$.
 For a $C^r$ diffeomorphism $f: U \to V$ between open subsets $U, V \subset {\mathbb R}^2$ and $r\in{\mathbb N}$, we set $\norm{D^rf} = \max_{x \in U,|{\alpha}|=r}\{\|\partial^{{\alpha}}f(x)\|\}$, $\norm{f}_{C^r} = \sup_{1 \leq \ell \leq r} \norm{D^\ell f}$ and $\norm{f}_{\diff^r(U)} = \norm{f}_{C^r(U)}+\norm{f^{-1}}_{C^r(V)}$.
 For $C^r$ diffeomorphisms $f,g: U \to V$, we denote $d_{\diff^r(U)}(f,g) = \max( \norm{f^{-1}g}_{\diff^r(U)}, \norm{g^{-1}f}_{\diff^r(U)})$. We abbreviate $\norm{f}_{\diff^r(U)}$, resp. $d_{\diff^r(U)}$, as $\norm{f}_{\diff^r}$, resp. $d_{\diff^r}$, when there is no confusion.
 For two homeomorphisms $f,g : U \to V$, we denote
$\norm{f-g} = \sup_{x \in U}d(f(x), g(x))$, where $d(\cdot,\cdot)$ is the euclidean metric.

\section{Preliminaries}\label{sec prelim}
We denote the boundary components of ${\mathbb A}$ by
 $$B_i := {\mathbb R} / {\mathbb Z} \times \{i\}\qquad\qquad i = 0,1.$$
The universal covering of ${\mathbb A}$ is $\tilde {\mathbb A} = {\mathbb R} \times [0,1]$.
We write $\tilde B_i = {\mathbb R}  \times \{i\}$ for $i=0,1$.
We let
 $$\pi : \tilde {\mathbb A} \to {\mathbb A}$$
be the natural projection, and let $T: \tilde {\mathbb A} \to \tilde {\mathbb A}$ be the translation in the first coordinate, i.e., $T(x,y) = (x + 1, y)$.
Given a homeomorphism $f:{\mathbb A}\to{\mathbb A}$ there is a lift
$F:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$ of $f$, also a homeomorphism, unique up to composition by a power of $T$, such that $\pi F = f \pi$.   Moreover, any such lift commutes with $T$.

We have the following lemma, whose proof is elementary and left to the readers.
\begin{lemma}\label{lem:lift}
	Suppose $f:{\mathbb A}\to{\mathbb A}$ is a homeomorphism with $\norm{f - {\rm Id}}<1/2$.   Then there is a unique lift $F:\tilde{\mathbb A}\to\tilde{\mathbb A}$ satisfying
	$\norm{F - {\rm Id}}<1/2$.   This lift satisfies
	\begin{eqnarray}\label{lift}
	d\big(F(\tilde x),\,\tilde x \big)\ =\ d\big(f(x),\,x\big)
	\end{eqnarray}
	for all $x\in{\mathbb A}$ and all $\tilde x\in\tilde{\mathbb A}$ with $\pi(\tilde x)=x$.
\end{lemma}

\begin{comment}
\begin{proof}
	We fix an arbitrary $x_0 \in {\mathbb A}$ and an arbitrary lift $\tilde x_0 \in \pi^{-1}(x_0)$.
	Let $F$ be an arbitrary lift of $f$ such that $d(F(\tilde x_0), \tilde x_0) = d(f(x_0), x_0)$. By definition, there is a function $k: {\mathbb A} \to {\mathbb Z}$ such that for any $x \in {\mathbb A}$, for any $\tilde x \in \pi^{-1}(x)$ we have
	\begin{eqnarray*}
	d(F(\tilde x), T^{k(x)}(\tilde x) ) = d(f(x), x).
	\end{eqnarray*}
	We claim that $k$ is locally constant. Indeed, for any $x \in {\mathbb A}$, any $\tilde x \in \pi^{-1}(x)$, and any $y \in {\mathbb A}$ sufficiently close to $x$, we let $\tilde y \in \pi^{-1}(y)$ be the element closest to $\tilde x$, and observe that
	\begin{eqnarray}
	d(T^{k(x)}(\tilde x), T^{k(y)}(\tilde y)) &\leq&
	d(F(\tilde x), T^{k(x)}(\tilde x) ) + d(F(\tilde y), T^{k(y)}(\tilde y) ) + d(F(\tilde x), F(\tilde y) )  \nonumber \\
	&=& d(f(x), x) + d(f(y), y) +   d(F(\tilde x), F(\tilde y)).  \label{eq upperboundforkxky}
	\end{eqnarray}
	As $y$ tends $x$, the right hand side of \eqref{eq upperboundforkxky} tends to $2d(f(x),x)$, which is strictly smaller than $1$ by our hypothesis. This is only possible if $k(y) = k(x)$ for all $y$ sufficiently close to $x$.

	Since ${\mathbb A}$ is connected and by definition $k(x_0) = 0$, we conclude that $k \equiv 0$. This concludes the proof.
\end{proof}

\end{comment}

\begin{defi}
	We say that $\gamma\subset{\mathbb A}$ is a {\it simple regular curve} connecting $B_0$ and $B_1$ if $\gamma = \phi([0,1])$ where $\phi : [0,1] \to {\mathbb A}$ is an injective continuous map  mapping $0$, resp. $1$, into $B_0$, resp. $B_1$, that maps $(0,1)$ into ${\mathbb A}\backslash\partial{\mathbb A}$.  	We define simple regular curves in $\tilde{\mathbb A}$ connecting $\tilde B_0$ and $\tilde B_1$ in a similar way.
\end{defi}

\begin{defi}\label{ordering on curves}
	For a pair of simple regular curves $\gamma_1,\gamma_2$ in $\tilde{\mathbb A}$ we will say that \textit{$\gamma_1$ is to the left of $\gamma_2$}, or \textit{$\gamma_2$ is to the right of $\gamma_1$}, and write
	\begin{eqnarray*}
					\gamma_1 < \gamma_2
	\end{eqnarray*}
	if $\gamma_1\cap\gamma_2=\emptyset$ and $\gamma_1$ lies in the component of $\tilde{\mathbb A}\backslash\gamma_2$ containing points with arbitrarily
	negative ${\mathbb R}$-coordinate.   This defines a partial ordering on simple regular curves in $\tilde{\mathbb A}$.
\end{defi}

\begin{defi}\label{def:Brouwer}
	Let $f$ be a homeomorphism of ${\mathbb A}$.
A simple regular curve $\gamma$ in ${\mathbb A}$ connecting $B_0$ and $B_1$, satisfying
$$\gamma\cap f(\gamma)=\emptyset$$
is called a \textit{Brouwer curve} for $f$. A Brouwer curve $\gamma$ is called \textit{smooth} if $\gamma$ is a $C^\infty$-curve.   Brouwer curves for homeomorphisms of $\tilde{\mathbb A}$ are defined in a similar way.

\end{defi}
Note that if $F$ is a lift of a homeomorphism $f:{\mathbb A}\to{\mathbb A}$ and $\tilde\gamma$ is a lift of a Brouwer curve $\gamma\subset{\mathbb A}$ for $f$, then
\begin{eqnarray*}
T^{k}F(\tilde \gamma) \cap \tilde\gamma = \emptyset \qquad \forall k \in {\mathbb Z}.
\end{eqnarray*}
\begin{defi}
	Let $f$ be a homeomorphism of ${\mathbb A}$ and $Q\geq 2$ an integer.  We say that $\gamma$ is a \textit{$Q$-good smooth curve} if $\gamma$ is a smooth Brouwer curve for each of the maps $f,f^2,\cdots,f^{Q-1}$.
\end{defi}

\begin{defi}
	Let $\gamma_1, \gamma_2$ be two disjoint simple regular curves in ${\mathbb A}$ connecting $B_0$ and $B_1$.    There is a unique closed region $\mathcal R$
	in ${\mathbb A}$ with left boundary $\gamma_1$ and right boundary $\gamma_2$.   More precisely, if each $\gamma_i$ is oriented from $B_0$ to $B_1$, then $\partial\mathcal R\cap ({\mathbb A}\backslash\partial{\mathbb A})$ has orientation agreeing with $\gamma_2 - \gamma_1$.  We say that \textit{$\mathcal R$ is the region bounded by $(\gamma_1, \gamma_2)$}.
\end{defi}

\begin{defi}
	Let $\{K_r\}_{r \geq 1}$ be an increasing sequence of positive real numbers. Let $U,V$ be smooth surfaces, possibly with boundary, and let $\phi: U \to V$ be a
	$C^\infty$-diffeomorphism. We say that \textit{$\phi$ is $\{K_r\}_{r \geq 1}$-smooth} if 	\begin{eqnarray*}
	\norm{\phi}_{\diff^r(U)} < K_r \qquad \forall r \in {\mathbb Z}_+.
	\end{eqnarray*}
\end{defi}
\begin{defi}\label{def ad coord}
	Let $\gamma$ be a Brouwer curve for $f$, and let $\mathcal R \subset {\mathbb A}$ be the  closed region  bounded by $(\gamma, f(\gamma))$. We say that an orientation  	preserving $C^\infty$ -diffeomorphism
	$$\phi : \mathcal U \to \mathcal R'$$
	from  an open neighborhood $\mathcal U\subset \tilde{{\mathbb A}}$ of $[0,1]^2$ to an open neighborhood $\mathcal R'\subset{\mathbb A}$ of $\mathcal R$ is
	an \textit{admissible coordinate} for $(\mathcal R, f)$ if the following hold:
	\begin{enumerate}
	\item $\phi$ has constant Jacobian,
	\item $\phi$ satisfies
	\begin{eqnarray}
	\phi(\{0\} \times [0,1]) &=& \gamma, \label{item 1} \\
	\phi(\{1\} \times [0,1]) &=& f(\gamma), \\
	\phi([0,1] \times \{0\}) &\subset& B_0, \\
	\phi([0,1] \times \{1\}) &\subset& B_1, \label{item 4}
	\end{eqnarray}
	\item there is a neighborhood $U_L$ of $\{0\} \times [0,1]$ in $\mathcal U$, so that
	\begin{eqnarray*}
	f \phi(x) = \phi T(x)  \qquad \forall x \in \mathcal U_L.
	\end{eqnarray*}
	\end{enumerate}
	Without loss of generality we may assume that $T(\mathcal U_L) \subset \mathcal U$ by choosing $U_L$ sufficiently small.

We make a similar definition for lifts.
Namely, let $\tilde \gamma$ be a lift of a Brouwer curve $\gamma$ for $f$,
let $F$ be an lift of $f$ such that
$F(\tilde \gamma)$ is on the right of $\tilde \gamma$,
and let $\tilde \mathcal R$ be the region bounded by $(\tilde \gamma, F(\tilde \gamma))$.    \color{black} We say that a $C^\infty$-diffeomorphism
$$\phi: \mathcal U \to \mathcal R'$$
from an open neighborhood $\mathcal U$ of $[0,1]^2$ in $\tilde{\mathbb A}$ to an open neighborhood $\mathcal R'$ of $\tilde \mathcal R$ in $\tilde{\mathbb A}$ is an \textit{admissible coordinate for $(\tilde \mathcal R, F)$} if $\phi$ satisfies the analogous properties above with $(\mathcal R, {\mathbb A}, f)$ replaced by $(\tilde \mathcal R, \tilde {\mathbb A}, F)$.
\end{defi}

\begin{rema} \label{rema unique continuation}
	We notice that by item $(3)$ in Definition \ref{def ad coord}, an admissible coordinate $\phi: \mathcal U \to \mathcal R'$ is determined by its restriction to $[0,1]^2$ in the following sense: if $\phi_i: \mathcal U_i \to \mathcal R_i$, $i=1,2$ are two admissible coordinates for $(\mathcal R, f)$ such that $\phi_1|_{[0,1]^2} = \phi_2|_{[0,1]^2}$, then there exists an open neighborhood $\mathcal U_3$ of $[0,1]^2$ in $\tilde{{\mathbb A}}$ such that
	$\phi_1|_{\mathcal U_3} = \phi_2|_{\mathcal U_3}$. For this reason, we will sometimes identify an admissible coordinate for $(\mathcal R, f)$ or $(\mathcal R, F)$ with a map  from $[0,1]^2$ to $\mathcal R$.

\end{rema}

\begin{defi}\label{def ad coord 2}
	Let $r\in{\mathbb N}$ and $K\in(0,\infty)$.
	We say that $\phi: [0,1]^2 \to \mathcal R$ is a $(r, K)$-{\it admissible coordinate for $(\mathcal R, f)$} if $\phi$ is an admissible coordinate, in the sense of
	Definition \ref{def ad coord}, defined on an open neighborhood $\mathcal U$ of $[0,1]^2$, satisfying
	\begin{eqnarray*}
	\norm{\phi}_{\diff^r(\mathcal U)} < K.
	\end{eqnarray*}
Analogously for lifts: We say that $\phi: \mathcal U \to \tilde\mathcal R$ is a $(r, K)$-{\it admissible coordinate for $(\tilde\mathcal R, F)$} if $\phi$ is an admissible coordinate
for $(\tilde\mathcal R, F)$, in the sense of Definition \ref{def ad coord}, defined on an open neighborhood $\mathcal U$ of $[0,1]^2$, satisfying $\norm{\phi}_{\diff^r(\mathcal U)} < K$.
\end{defi}

\begin{comment}
\blue{

\begin{lemma}\label{extension}
If $\phi:\mathcal U \to \tilde \mathcal R$ is an admissible coordinate for $(\tilde \mathcal R, F)$, then it extends uniquely to a $C^r$-diffeomorphism
\begin{eqnarray*}
			\bar\phi:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}
\end{eqnarray*}
with constant Jacobian, satisfying
\begin{eqnarray}\label{extension commutes}
					\bar\phi\circ T=F\circ\bar\phi.
\end{eqnarray}
If moreover $\phi$ is an $(r,K)$-admissible coordinate for $(\tilde \mathcal R, F)$, then the extension satisfies
\begin{eqnarray}\label{estimates on [j,j+1]}
				\qquad	\norm{\bar\phi}_{\diff^r([j,j+1]\times[0,1])} < M_r(K+K^r)\Big(\norm{F^j}_{\diff^r(\tilde{\mathbb A})} + \norm{F^j}^r_{\diff^r(\tilde{\mathbb A})}\Big)
\end{eqnarray}
for each $j\in{\mathbb Z}$, for some universal constant $M_r>0$ depending only on $r\in{\mathbb N}$.
\end{lemma}
\begin{proof}
For each $x\in\tilde{\mathbb A}$ there is a unique $j\in{\mathbb Z}$ so that $T^jx\in \tilde\gamma\cup\textup{int}(\tilde\mathcal R)$.     Setting $\bar\phi(x):=F^{-j}\circ\phi\circ T^j(x)$
we see that $\bar\phi$ is clearly as smooth as $\phi$ away from the $T$-translates of $\tilde\gamma$.    By condition (3) in Definition \ref{def ad coord}
the map $\bar\phi$ is moreover as smooth as $\phi$ at each of the $T$-translates of $\tilde\gamma$.

It remains to show (\ref{estimates on [j,j+1]}).    We will use the following classical estimate: for $C^r$-maps $g:U\rightarrow V$ and $f:V\rightarrow W$,
\begin{eqnarray*}
		\quad	\|f\circ g\|_{\diff^r(U)}\leq M_r \big(\|f\|_{\diff^r(V)}+\|f\|^r_{\diff^r(V)} \big)\big(\|g\|_{\diff^r(U)} +\|g\|^r_{\diff^r(U)}\big)
\end{eqnarray*}
for some constant $M_r>0$ independent of $f$ and $g$, provided the dimensions of $U,V, W$ are fixed.
Now fix $j\in{\mathbb Z}$.    Then by (\ref{extension commutes}) $\bar\phi=F^j\circ\bar\phi\circ T^j$
on $\tilde{\mathbb A}$.   So,
\begin{eqnarray*}
				\norm{\bar\phi}_{\diff^r([j,j+1]\times[0,1])} &=& \norm{F^j\circ\bar\phi\circ T^j}_{\diff^r([j,j+1]\times[0,1])} \\
											&=& \norm{F^j\circ\bar\phi}_{\diff^r([0,1]\times[0,1])} \\
											&=& \norm{F^j\circ\phi}_{\diff^r([0,1]\times[0,1])} \\
											&\leq & M_r\big(\norm{F^j}_{\diff^r(\tilde{\mathbb A})}+\norm{F^j}^r_{\diff^r(\tilde{\mathbb A})} \big)\cdot\\
											&  &\qquad\quad \big( \norm{\phi}_{C^r([0,1]\times[0,1])} + \norm{\phi}^r_{C^r([0,1]\times[0,1])}\big)\\
											&= & M_r\big(\norm{F^j}_{\diff^r(\tilde{\mathbb A})}+\norm{F^j}^r_{\diff^r(\tilde{\mathbb A})} \big)
																		\big( K + K^r\big)
\end{eqnarray*}
as claimed.
\end{proof}

\end{comment}

\begin{defi} 	\label{defi prot}
A  \textit{pseudo-rotation} is a non-wandering homeomorphism $f : {\mathbb A} \to {\mathbb A}$ that is isotopic to the identity, maps $B_0$ (resp.\ $B_1$) to itself,  and has no periodic points.
\end{defi}
Recall that a homeomorphism $f: {\mathbb A} \to {\mathbb A}$ is said to be {\it non-wandering} if for every open subet $U \subset {\mathbb A}$, there exists an integer $n > 0$ such that $f^n(U) \cap U \neq \emptyset$.
In this paper we will only be considering $C^\infty$-smooth pseudo-rotations that preserve a smooth area form $\omega$.
By a slight abuse of notation, we denote by $\omega$ both the area form on $\tilde {\mathbb A}$ and ${\mathbb A}$.
If $f$ is an $\omega$-preserving diffeomorphism then so is each lift $F$ an $\omega$-preserving diffeomorphism.

\begin{defi}\label{def rotation number}
	Let $f: {\mathbb A} \to {\mathbb A}$ be a pseudo-rotation with lift $F:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$. Denote by $p_1 : \tilde {\mathbb A} \to {\mathbb R}$ the projection on the first coordinate. We define $\rho(F) \in {\mathbb R}$ by
	\begin{eqnarray*}
	\rho(F) := \lim_{n \to \infty} \frac{1}{n}(p_1(F^{n}(x,y)) - x ) \qquad x \in {\mathbb R}, y \in [0,1].
	\end{eqnarray*}
	It is known that the limit on the right hand side above exists for all $(x,y)$ and is independent of $(x,y)$, see \cite{F88, FH12}.  Moreover, we always have $\rho(F) \notin {\mathbb Q}$, see \cite{F88}.
For any pseudo-rotation $f$, there exists a unique lift $F$ of $f$ such that $\rho(F) \in (0,1)\setminus {\mathbb Q}$, which we denote by $\rho(f)$.   We call $\rho(f)$ the
\textit{rotation number} of $f$.
\end{defi}

Let us  recall a few well known facts about the best rational approximations to ${\alpha}\in (0,1)\setminus {\mathbb Q}$ and its continued fraction expansion.   Readers can consult \cite[Chapters X, XI]{HW} for more details.

First, for $x\in{\mathbb R}$ we will write
\begin{eqnarray*}
		\lfloor x\rfloor &:= &\max\{ n\in{\mathbb Z}\ |\ n\leq x\} \in {\mathbb Z}, \\
		q(x) &:= &\lfloor 1/x\rfloor \in {\mathbb Z}
\end{eqnarray*}
for the integer parts of $x$ and $1/x$ respectively.    Moreover, if $x\in {\mathbb R}\setminus {\mathbb Q}$ then
\begin{eqnarray*}
			\norm{x}_{{\mathbb R} / {\mathbb Z}} := d(x, {\mathbb Z})\in (0,1/2).
\end{eqnarray*}
The Gauss map ${\mathcal G}: [0,1)\rightarrow[0,1)$ is defined by
\begin{eqnarray*}
			{\mathcal G}(x)=\frac{1}{x} - q(x)
\end{eqnarray*}
on $(0,1)$ and ${\mathcal G}(0):=0$.   If $x$ is irrational then so is ${\mathcal G}(x)$.

\begin{defi}
	For any ${\alpha} \in (0,1) \setminus {\mathbb Q}$, we define the sequences $({\alpha}_n)_{n \geq 0}$ and $(\beta_n)_{n\geq0}$ in $(0,1) \setminus {\mathbb Q}$ by
	\begin{eqnarray*}
	{\alpha}_0 := {\alpha}, \qquad {\alpha}_n &:=&           \mathcal G^n({\alpha}_0)\qquad \forall n \geq 1,\\
	\beta_n &:=&\prod_{i=0}^{n}{\alpha}_i \qquad \forall n \geq 0.
	\end{eqnarray*}
	Furthermore we define sequences of non-negative integers $(a_n)_{n \geq 0}$, $(q_n)_{n \geq 0}$  as follows:
	\begin{eqnarray}
	a_0 := 0, \qquad  \quad a_n &:=& q({\alpha}_{n-1})\qquad \forall n \geq 1,  \label{eq a n 0}\\
	q_{0}:=1,\quad q_1:=q(\alpha), \quad  q_{n+2} &:=& q_n + q_{n+1} a_{n+2} \qquad \forall n \geq 1.  \label{q iteration}

	\end{eqnarray}
	We also define $(p_n)_{n \geq 0}$ by $p_0 = 0$, and for $n \geq 1$, define
	 $p_n$ to be the closest integer to $q_n\alpha$, which is unique by irrationality of $\alpha$. We will use the notation ${\alpha}_n({\alpha})$, $q_n({\alpha})$ and $p_n({\alpha})$ when it is necessary to indicate the dependence of the sequences on ${\alpha}$.
\end{defi}
	Note that
	\begin{eqnarray} \label{eq a n}
	 \qquad {\alpha}_{n-1}^{-1} = a_n + {\alpha}_n  \qquad\forall n \geq 1,
	\end{eqnarray}
	since ${\alpha}_n=          \mathcal G({\alpha}_{n-1})=1/{\alpha}_{n-1}-q({\alpha}_{n-1})=1/{\alpha}_{n-1}-a_n$.
	It is also well known that
	\begin{eqnarray}
		p_{n+1}q_n-p_nq_{n+1}&=& (-1)^n \label{relation pn qn}\\
	  \beta_n = (-1)^n(q_n {\alpha} - p_n) &>& 0\qquad\forall n \geq 0.\label{sign}
	\end{eqnarray}
	In particular for each $n \geq 1$, $\beta_n=|q_n{\alpha} - p_n| =d(q_n{\alpha},{\mathbb Z})$, and
	\begin{eqnarray*}
			{\alpha}
				\xcontfrac{;a_1,a_2,\cfracdots,a_n + {\alpha}_n}
			\qquad\qquad
			\frac{p_n}{q_n}
				\xcontfrac{;a_1,a_2,\cfracdots,a_n}
	\end{eqnarray*}
	For any $n \geq 1$,
	 the integers $p_n$ and $q_n$ are relatively prime, and
	$p_n / q_n$ is called the $n$-th best rational approximation of ${\alpha}$.   The following simple inequalities are known, and will be used later:
	\begin{eqnarray} \label{eq continuefraction}
	 \frac{1}{2q_{n+1}} < \frac{1}{q_n + q_{n+1}} <           \beta_n < \frac{1}{q_{n+1}}, \\
	\label{eq anupperlower}
	 {\alpha}_n,\  q({\alpha}_n)^{-1} \in \left(\frac{q_n}{2q_{n+1}}, \frac{2q_n}{q_{n+1}}\right).
	\end{eqnarray}

\section{Existence of a Brouwer curve with uniform bounds}

In order to avoid lengthy computations, in the rest of the paper we will introduce various {\it increasing functions} to keep track of parameter dependence.  We say that a function $A : U \to {\mathbb R}$ defined on an open subset $U \subset {\mathbb R}^n$ is {\it increasing} if for any $x = (x_1, \cdots, x_n)$ and $y = (y_1, \cdots, y_n) \in U$ with $x_i \geq y_i$ for all $1 \leq i \leq n$, we have $A(x) \geq A(y)$. We define  {\it decreasing functions} in a similar way. Typically, all the variables and values of the increasing/decreasing functions that we will consider lie in ${\mathbb R}_+$.

The following theorem, essentially proven in \cite{A}, allows one to control the $C^r$-distance of a pseudo-rotation to the identity in terms of its rotation number.

\begin{thm}\label{thm aflxz}
	There is a sequence of  increasing functions
	\begin{eqnarray*}
				&  &A_r : (0, 1/2) \times  {\mathbb R}_+ \to {\mathbb R}_+, \\
				&	&\lim_{t \to 0}A_r(t,\, \cdot\,) \equiv 0
	\end{eqnarray*}
	for each $r\in{\mathbb N}$, so that for any $\{ K_r \}_{r \geq 1}$-smooth $\omega$-preserving pseudo-rotation $f:{\mathbb A}\to{\mathbb A}$ there holds
	\begin{eqnarray*}
	\norm{f - {\rm Id}} &<& A_0(\norm{\rho(f)}_{{\mathbb R} / {\mathbb Z}},  K_1),\\
	\norm{D^rf} &<&  A_r(\norm{\rho(f)}_{{\mathbb R} / {\mathbb Z}}, K_{r+1}), \quad \forall r \geq 1.
	\end{eqnarray*}
\end{thm}
\begin{proof}
	By \cite[Corollary A]{A}, we know that
	\begin{eqnarray*}
	\norm{f-{\rm Id}} < (1+2K_1)\norm{\rho(f)}_{{\mathbb R} / {\mathbb Z}}^{1/2} =: A_0(\norm{\rho(f)}_{{\mathbb R} / {\mathbb Z}},K_1).
	\end{eqnarray*}
	By the Hadamard-Kolmogorov convexity theorem, we get corresponding bounds on the ``inbetween'' derivatives: for any $r \geq 1$
	\begin{eqnarray*}
	\norm{D^r f} &\leq& C_r \norm{f-{\rm Id}}^{\frac{1}{r+1}}\norm{f}_{\diff^{r+1}}^{\frac{r}{r+1}} \\
	&\leq&C_r \norm{\rho(f)}_{{\mathbb R} / {\mathbb Z}}^{\frac{1}{2(r+1)}}(1+2K_{r+1}) =: A_r( \norm{\rho(f)}_{{\mathbb R} / {\mathbb Z}},  K_{r+1}).
	\end{eqnarray*}
\end{proof}

By Theorem \ref{thm aflxz}, there is a decreasing function
\begin{eqnarray}  \label{def a}
\rho_* :{\mathbb R}_+ \to (0,1/2)
\end{eqnarray}
such that for any smooth  $\omega$-preserving pseudo-rotation $f$ satisfying
\begin{eqnarray}\label{lem:lift2}
\|\rho(f)\|_{{\mathbb R} / {\mathbb Z}} < \rho_*(\norm{Df}),
\end{eqnarray}
we have $$\norm{f - {\rm Id}} \leq A_0(\|\rho(f)\|_{{\mathbb R}/{\mathbb Z}},\|Df\|) < A_0(\rho_*(\|Df\|),\|Df\| )< 1/2.$$
Then by Lemma \ref{lem:lift}, there is a unique lift $F$ of $f$ satisfying
\begin{eqnarray} \label{lem Fidfid}
\norm{F - {\rm Id}} = \norm{f - {\rm Id}}  <1/2.
\end{eqnarray}

\begin{comment}
   \color{red}

The conclusions in Theorem \ref{thm aflxz} can be extended to lifts using the following:

   \color{black}

	\begin{lemma}\label{lem:lifts1}
		There exists a \blue{decreasing} function
		\begin{eqnarray*}
					a:{\mathbb R}_+ \to (0,1/2)
		\end{eqnarray*}
		so that the following holds.   If $f$ is any smooth  $\omega$-preserving pseudo-rotation satisfying
		\begin{eqnarray}\label{lem:lift2}
					\|\rho(f)\|_{{\mathbb R} / {\mathbb Z}} < a(\norm{Df}),
		\end{eqnarray}
		then
		\begin{eqnarray}\label{lift2}
		\norm{F(\tilde x) - \tilde x}=\norm{f(x) - x} < 1/2
		\end{eqnarray}
		for all $x\in{\mathbb A}$ and all $\tilde x\in\tilde{\mathbb A}$ with $x=\pi(\tilde x)$, where
		$F:\tilde{\mathbb A}\to\tilde{\mathbb A}$ is the unique lift of $f$ for which $\rho(F) \in (-1/2, 1/2)$.
\end{lemma}

\begin{proof}
Let $A_0$ be the function given by Theorem \ref{thm aflxz}.
For each $K>0$ we set
\begin{eqnarray*}
a(K) = \frac{1}{2}\sup \{ t \in (0,1/4]  \mid   A_0(t,K) < 1/2 \}.
\end{eqnarray*}
Since for each $K > 0$ we have $\lim_{t \to 0}A_0(t, K) = 0$, so is $a(K)$ well defined and positive, and by construction
$A_0(a(K),K)<1/2$.   \blue{Moreover $a$ is decreasing.}
    If $f$ satisfies (\ref{lem:lift2}), then
$$\norm{f - {\rm Id}} < A_0(\|\rho(f)\|_{{\mathbb R}/{\mathbb Z}},\|Df\|) < A_0(a(\|Df\|),\|Df\| )< 1/2.$$
Therefore, by Lemma \ref{lem:lift}, there is a unique lift $F$ of $f$ satisfying
$$\norm{F - {\rm Id}}<1/2,$$
and for this lift (\ref{lift2}) holds.  It is direct to see that $F$ is the unique lift of $f$ with rotation number in $(-1/2, 1/2)$.
\end{proof}
   \color{black}

\end{comment}

\begin{lemma}\label{lem no almost fixed point}
		There are increasing functions
		$$C', C'' : {\mathbb R}_+\times(0,1) \to {\mathbb R}_+$$
		with $\lim_{t \to 0} C''(t,\cdot)\equiv 0$ and
		\begin{eqnarray} \label{eq c''issmall}
		C''(\alpha, K^{-1}) < \rho_*(K^{\alpha^{-1}}) \quad \forall (\alpha,K) \in (0,1)\times(1,\infty)
		\end{eqnarray}
		such that the following holds:  if $\alpha\in (0,1)\backslash{\mathbb Q}$ satisfies
		\begin{eqnarray}
		{\alpha} &\in& (0,\rho_*(K)), \label{eq no fixed 02} \\
			{\mathcal G}({\alpha}) &<& C''({\alpha}, K^{-1}) \label{eq no fixed 01}
		\end{eqnarray}
		for some $K>1$,
		where $\rho_*$ is given in \eqref{def a}, then every $C^1$-smooth $\omega$-preserving pseudo-rotation with $\norm{Df} < K$
		and $\rho(f)={\alpha}$ satisfies
		\begin{eqnarray}\label{min distance}
		\inf_{x \in {\mathbb A}} d(x, f(x)) \geq C'({\alpha}, K^{-1} ).
		\end{eqnarray}
	\end{lemma}

	\begin{proof}
		Fix $K>0$ and ${\alpha}\in(0,1)\backslash{\mathbb Q}$ satisfying (\ref{eq no fixed 02}) and (\ref{eq no fixed 01}).   Let $f$ be a $C^1$-smooth
		$\omega$-preserving pseudo-rotation with $\norm{Df} < K$ and $\rho(f)={\alpha}$.
		Denote by $F$ the lift of $f$ with $\rho(F) = \rho(f)$.  We abbrieviate $q := q({\alpha})=\lfloor 1/{\alpha}\rfloor$.

		Choose the function $C'' :  {\mathbb R}_+ \times(0,1) \to {\mathbb R}_+$, with $\lim_{t \to 0}C''(t,\cdot)\equiv 0$, sufficiently small so that for each $t\in(0,1)$,
		the condition
		\begin{eqnarray*}
				s < C''(t, K^{-1})
		\end{eqnarray*}
		implies
		\begin{eqnarray}\label{eq no fixed 1}
					A_0(s, K^{t^{-1}})<1/2.
		\end{eqnarray}
		Without loss of generality we may arrange that $C''$ is increasing and satisfies \eqref{eq c''issmall}.    By Theorem \ref{thm aflxz}, we have
		\begin{eqnarray*}
		\|f^{q}-{\rm{Id}}\| \leq  A_0(\|\rho(f^{q})\|_{{\mathbb R}/{\mathbb Z}}, \|Df^{q}\|)
									\leq  A_0(\|q\rho(f)\|_{{\mathbb R}/{\mathbb Z}}, K^{q}).\label{bound on f^q}
		\end{eqnarray*}
		Since $\|q\rho(f)\|_{{\mathbb R}/{\mathbb Z}}=1-q{\alpha}={\alpha}{\mathcal G}({\alpha})\leq{\mathcal G}({\alpha})$ and $K^q\leq K^{{\alpha}^{-1}}$, the above gives
		\begin{eqnarray*}
		 \|f^{q}-{\rm{Id}}\| \leq  A_0({\mathcal G}({\alpha}), K^{{\alpha}^{-1}}).
		\end{eqnarray*}
		Thus, if ${\alpha}$ satisfies (\ref{eq no fixed 01}), then by (\ref{eq no fixed 1}),
		\begin{eqnarray} \label{eq no fixed 3}
		 \|f^{q}-{\rm{Id}}\|  <1/2.
		\end{eqnarray}
		The unique lift of $f^q$ with rotation number in $(-1/2, 1/2)$ is $T^{-1}F^q$, since
		$\rho(T^{-1}F^q)=q\rho(F)-1=q{\alpha}-1\in(-1/2,1/2)$.   So by Lemma \ref{lem:lift}
		\begin{eqnarray*}
		\|F^{q}-T\|= \|T^{-1}F^{q}-{\rm{Id}}\|=\|f^{q}-{\rm{Id}}\|.
		\end{eqnarray*}
		Thus 	for all $\tilde x\in\tilde{\mathbb A}$
		\begin{eqnarray} \label{eq no fixed 4}
		\qquad d(F^{q}(\tilde x),\tilde x)\geq d(T(\tilde x),\tilde x) - d(F^{q}(\tilde x),T(\tilde x)) > 1-1/2 = 1/2.
		\end{eqnarray}
	   On the other hand, for each $\tilde x\in\tilde{\mathbb A}$, there holds
		\begin{eqnarray*}
		d(F^q(\tilde x),\tilde x)  \leq  \sum_{j=0}^{q-1}d\big(F^{j+1}(\tilde x),F^j(\tilde x)\big)
							\leq  \sum_{j=0}^{q-1}\|DF\|^jd\big(F(\tilde x),\tilde x\big) \leq Cd(F(\tilde x),\tilde x)
		\end{eqnarray*}
		where $C=\frac{K^{q}-1}{K-1}\leq \frac{K^{{\alpha}^{-1}}-1}{K-1}$.
		Combining the last line with (\ref{eq no fixed 4}) and \eqref{lem Fidfid}, we obtain
		\begin{eqnarray}\label{eq no fixed 5}
					\|F(\tilde x)-\tilde x\| \geq \frac{1}{2C}\qquad \forall \tilde x\in\tilde{\mathbb A}.
		\end{eqnarray}
	    By Lemma \ref{lem:lift} and \eqref{eq no fixed 02}, we conclude that (\ref{min distance}) holds if we define $C'$ by
		\begin{eqnarray*}
						C'(t,K^{-1}) = \frac{1}{2}\frac{K-1}{K^{t^{-1}}-1}.
		\end{eqnarray*}
	\end{proof}

	\begin{defi}\label{def L(K)}
		For each $K>1$ let $\mathcal L(K)$ be the set of ${\alpha} \in (0, 1/2) \setminus {\mathbb Q}$
		such that
		\begin{eqnarray}
		\label{eq def cL3}
		{\alpha} &\in &  \big(0,\rho_*(K)\big), \\
		\label{eq def cL2}
		          \mathcal G({\alpha}) &<& C''({\alpha}, K^{-1})
		\end{eqnarray}
		where $\rho_*$ is from \eqref{def a} and $C''$ is from Lemma \ref{lem no almost fixed point}.
	\end{defi}

\begin{lemma}\label{lem brouwer curve compactness}
	There is a sequence of  increasing functions
	$$C'_r : {\mathbb R}_+^{2} \to {\mathbb R}_+\qquad\forall r\geq 1,$$
	such that  the following is true.   If $\{K_r\}_{r\geq 1}$ is any sequence in $(1,\infty)$ and ${\alpha}\in \mathcal L(K_1)$, then every
	$\{K_r\}_{r \geq 1}$-smooth $\omega$-preserving pseudo-rotation $f$ with $\rho(f)={\alpha}$ has for each $r\in{\mathbb N}$
	a Brouwer curve $\gamma=\gamma_r$ for which
	the closed region in ${\mathbb A}$ bounded by $(\gamma, f(\gamma))$ admits a
	$(r, C'_r({\alpha}^{-1}, K_{r+1}) )$-admissible coordinate.
\end{lemma}

\begin{proof}
	Fix a sequence $(K_r)_{r\geq 1}$ in $(1,\infty)$.   Fix $r\geq 1$.
	For each $c>0$ and $K\geq 1$ define the following subset of $\diff^{r}({\mathbb A}, \omega)$:
	\begin{eqnarray*}
 			          \mathcal H_{r}(c,K) := \Big\{g \in \diff^{r}({\mathbb A}, \omega)\ \Big|\ \norm{g}_{\diff^{r}} \leq K,\quad \inf_{x \in {\mathbb A}} d(x, g(x)) \geq c \Big\}.
	\end{eqnarray*}
Denote by
\begin{eqnarray*}
	\overline{          \mathcal H_{r+1}(c,K)} \subset \diff^{r}({\mathbb A}, \omega)
\end{eqnarray*}
the closure of $          \mathcal H_{r+1}(c,K)$ in the $C^r$-topology.   We observe that $\overline{          \mathcal H_{r+1}(c,K)}$ is compact in $\diff^{r}({\mathbb A})$ and contains only diffeomorphisms without fixed points.
Moreover, for $c'\geq c$ and $K'\leq K$ we have $\overline{          \mathcal H_{r+1}(c',K')}\subset \overline{          \mathcal H_{r+1}(c,K)}$.

We now prove a version of the lemma for elements of $\overline{          \mathcal H_{r+1}(c,K)}$, for each fixed $c>0, K\geq1$.   Then we argue that the union of
$\overline{          \mathcal H_{r+1}(c,K_{r+1})}$ over $c\in(0,1]$ contains all pseudo-rotations satisfying the assumptions of the Lemma.

To this end, fix $c>0, K\geq1$ and consider $g \in \overline{          \mathcal H_{r+1}(c,K)}$.   Since $g$ has no fixed points, a strong refinement of Brouwer's plane
	translation theorem due to Guillou \cite[Th\'eor\`em 5.1]{G}
		yields a $C^0$ Brouwer curve $\gamma_0$
	for $g$ in the sense of Definition \ref{def:Brouwer}.
	Any sufficiently $C^0$-close smooth
	approximation of $\gamma_0$ that continues to connect the two boundary components yields a smooth Brouwer curve $\gamma$ for $g$.
	Clearly we can choose such a $\gamma$ to meet both boundary components of the annulus  orthogonally.   We can then apply the following lemma.

	\begin{lemma}\label{A:admissible-coordinates}
		Let $r \geq 1$ and $\theta \in (0,1)$. Suppose $g\in\diff^{r, \theta}({\mathbb A}, \omega)$ has a smooth Brouwer curve $\gamma$ that meets both boundary components  orthogonally.
		Then there exist $D = D(g, \gamma, r) > 1$, and a neighborhood $\mathcal V$ of $g$  in $\diff^{r, \theta}({\mathbb A}, \omega)$   such that for every  $g' \in \mathcal V$,
		the region $(\gamma,\,g'(\gamma))$ has
		admissible coordinates $\phi'$ whose $C^r$-norm is bounded by $D$.
	\end{lemma}
\begin{proof}
	 Let $\mathcal R$ denote the region $(\gamma,\,g(\gamma))$.   We first construct a $C^{\infty}$-diffeomorphism $\psi_L$ with constant Jacobian $\omega(\mathcal R)$ from a neighborhood of $\{0\} \times  [0,1]$ in $\tilde{\mathbb A}$ onto its image a neighborhood of $\gamma$ in ${\mathbb A}$.
	 Indeed, without loss of generality $\gamma:[0,1]\rightarrow{\mathbb A}$ meets the boundary of ${\mathbb A}$ orthogonally near both end points and
	 is parametrised by arclength so that $\|\dot{\gamma}\|=L$ is constant, $L\geq 1$.   Then $n:=-i\dot{\gamma}/L$ is a normal vector field along
	 $\gamma$ and the map $\tilde{A}\rightarrow{\mathbb R}^2,$ $(x,y)\mapsto \gamma(y)+x\omega(\mathcal R)n(y)/L$ extends $\gamma$ to a smooth diffeomorphism from a sufficiently small tubular neighborhood of $\{0\} \times  [0,1]$ in $\tilde{\mathbb A}$
	 to a neighborhood of the image of $\gamma$ in ${\mathbb A}$, with constant Jacobian $\omega(\mathcal R)$ along $\{0\} \times  [0,1]$ and also near to the boundary of $\tilde{\mathbb A}$.   The
	 Jacobian of this map away from $\{0\} \times  [0,1]$ depends only on the $y$ variable and therefore by a further change of coordinates it is easily modified to have
	 constant Jacobian on a whole neighborhood of $\{0\} \times  [0,1]$ while still mapping $\{0\} \times  [0,1]$ to $\gamma$.   The resulting map, which we denote by $\psi_L$, clearly has finite $C^{r, \theta}$-norm.

	 Then $g \psi_L T^{-1}$ is a $C^{r, \theta}$-diffeomorphism $\psi_R$ with constant Jacobian $\omega(\mathcal R)$ from a neighborhood of $\{1\} \times  [0,1]$ in $\tilde{\mathbb A}$ onto its image, a
	 neighborhood of $g(\gamma)$ in ${\mathbb A}$. We construct the chart $\phi: [0,1]^2 \to {\mathbb A}$ by first specifying its restriction to a neighborhood of $\{0\} \times [0,1]$, resp.\ $\{1\} \times [0,1]$, to be $\psi_L$, resp.\ $\psi_R$.   Then we extend it by hand to neighborhoods of $[0,1]\times\{0\}$ and $[0,1]\times\{1\}$, also with
	 constant Jacobian $\omega(\mathcal R)$.   Finally, using \cite[Corollary 4]{A10} (or, in our simple application, using directly \cite[Theorem 3]{A10} which follows from \cite{DacMos}), we extend the map to all of $[0,1]^2$ so as to have constant Jacobian,
	 and the $C^r$ norm of the resulting map is bounded in terms of the $C^{r, \theta}$ norm of $g$.
	   By construction the conditions in Definition \ref{def ad coord} hold.

	  Each $g$ has a $C^0$-small (and hence also $C^r$-small) neighborhood in $\diff^r({\mathbb A}, \omega)$ for which
	the same $\gamma$ can be applied.
	 The uniform bounds for the $C^r$-norm of $\phi$ is an immediate consequence of the construction.
	We omit the proof of this latter fact and refer the readers to \cite{A10, DacMos} for details.
\end{proof}

	By Lemma \ref{A:admissible-coordinates} and by compactness of $\overline{          \mathcal H_{r+1}(c,K)}\subset \diff^{r, 1/2}({\mathbb A}, \omega) \subset \diff^{r}({\mathbb A}, \omega)$, we find a finite collection of neighborhoods (with respect to the $C^{r, 1/2}$-topology) as in Lemma \ref{A:admissible-coordinates}
	whose union covers $\overline{          \mathcal H_{r+1}(c,K)}$, and thus obtain a uniform bound $E_r(c,K)>0$ on the $C^r$-norm of admissible coordinates that
	applies to all elements of $\overline{          \mathcal H_{r+1}(c,K)}$.
	Due to the inclusions $\overline{          \mathcal H_{r+1}(c',K')}\subset \overline{          \mathcal H_{r+1}(c,K)}$ for $c'>c, K'<K$, we can assume that $E_r(c,K)$ is
	decreasing in $c$ and increasing in $K$.

	Now, suppose $f$ is as in Lemma \ref{lem brouwer curve compactness}.  That is, $f$ is a $(K_r)_{r \geq 1}$-smooth $\omega$-preserving pseudo-rotation with
	rotation number $\rho(f)={\alpha}\in\mathcal L(K_1)$.    By Lemma \ref{lem no almost fixed point},
\begin{eqnarray*}
 			f\in           \mathcal H_{r+1}\Big(C'({\alpha}, K_1^{-1}),\ K_{r+1}\Big).
\end{eqnarray*}
Thus there exists a region $(\gamma,\,f(\gamma))$ having admissible coordinates whose
	$C^r$-norm is bounded by $E_r(C'({\alpha}, K_1^{-1}), K_{r+1})$, for some smooth Brouwer curve $\gamma$.  Hence Lemma \ref{lem brouwer curve compactness}
	holds with $C'_r({\alpha}^{-1},K_{r+1}):=E_r(C'({\alpha}, K_1^{-1}),K_{r+1})$.  Evidently $C'_r$ is an increasing function, since $C'$ is increasing and $E_r(c,K)$ is decreasing in $c$ and increasing in $K$.
\end{proof}

It will be useful to fix the following notation:

\begin{defi}\label{def Lr}
 For each integer $r\geq 1$ and each $K>1$ let $\mathcal L_r(K)$ be the set of ${\alpha} \in \mathcal L(K)$
such that
	\begin{eqnarray}
	\label{eq def cL}
	A_0({\alpha}          \mathcal G({\alpha}), K^{q({\alpha})})
	< K^{-q({\alpha})} C'_r({\alpha}^{-1},   K)^{-1}
	\end{eqnarray}
	where $C'_r : {\mathbb R}_+^{2} \to {\mathbb R}_+$, $r \geq 1$ are the increasing functions produced by
	Lemma \ref{lem brouwer curve compactness}.
\end{defi}

We will see later that each $\mathcal L_r( K)$ is non-empty for each integer $r \geq 1$ and each $K > 1$.  Moreover, by  \eqref{eq def cL2} and \eqref{eq def cL} we have
\begin{eqnarray}\label{cL nested}
\mathcal L_r(  K')\subset \mathcal L_r(K)\qquad\qquad \forall K'>K.
\end{eqnarray}

Now we can strengthen Lemma \ref{lem brouwer curve compactness}.  Indeed, we show that if more restrictions are placed on the rotation number of a $\omega$-preserving pseudo-rotation $f$, then the produced Brouwer curve
is actually $Q$-good, for some large $Q$ depending on the rotation number of $f$:

\begin{prop} \label{prop. main}
	Let $f$ be a $\{K_r\}_{r \geq 1}$-smooth $\omega$-preserving pseudo-rotation.
	If $\rho(f) \in \mathcal L_r( K_{r+1})$ for some integer $r\geq 1$, then the Brouwer curve $\gamma$ for $f$ produced by Lemma \ref{lem brouwer curve compactness} (corresponding to $r$)
	is $q(\rho(f))$-good.
\end{prop}

Note that the condition $\rho(f) \in \mathcal L_r( K_{r+1})$ in this Proposition includes the condition on $\rho(f)$ used in Lemma  \ref{lem brouwer curve compactness},
since $\mathcal L_r( K)\subset\mathcal L(K)$ by definition.

\begin{proof}

We denote by $d_H$ the Hausdorff distance on $\tilde{\mathbb A}$, i.e.\ for any two subsets $A,B$ of $\tilde{\mathbb A}$,
\begin{eqnarray*}
d_H(A,B) = \sup_{x \in A, y \in B} \max( d(x, B), d(y,A)).
\end{eqnarray*}

Set ${\alpha} = \rho(f) \in \mathcal L_r( K_{r+1})$ for some $r \geq 1$.
 Let $\gamma$ be the Brouwer curve given by Lemma \ref{lem brouwer curve compactness} for this value of $r$.
Let $\mathcal R\subset{\mathbb A}$ be the region bounded by $(\gamma, f(\gamma))$ and let $\psi:\mathcal R\rightarrow[0,1]\times[0,1]$ be the inverse of some $(r,  C'_r( {\alpha}^{-1}, K_{r+1}))$-admissible coordinates produced by
Lemma \ref{lem brouwer curve compactness}.    Applying the intermediate value theorem to $\psi$ yields
\begin{eqnarray} \label{eq dhgammafgamma}
d_H(\gamma, f(\gamma)) > C'_r( {\alpha}^{-1}, K_{r+1})^{-1}.
\end{eqnarray}

Fix a lift $\tilde\gamma\subset\tilde{\mathbb A}$ of $\gamma$, and let $F$ be the lift of $f$ for which $\rho(F) = {\alpha}$.   Since $\tilde\gamma\cap F(\tilde \gamma)=\emptyset$ it follows
from ${\alpha} > 0$ and the order of boundary points that $\tilde\gamma<F(\tilde \gamma)$.
By  Definition \ref{def ad coord}, \eqref{eq dhgammafgamma} and $K_{r+1} \geq K_1$, we can see that
\begin{eqnarray}
d_{H}( F^{q({\alpha})-1}(\tilde \gamma), F^{q({\alpha})}(\tilde \gamma) )
&>&   K_1^{-q({\alpha})}    \color{black} d_{H}( \tilde \gamma, F(\tilde \gamma) )  \nonumber \\
&\geq&  K_{r+1}^{-q({\alpha})}C'_r({\alpha}^{-1}, K_{r+1})^{-1}.  \label{eq def c0}
\end{eqnarray}
Here we implicitely used that the analogue of \eqref{eq dhgammafgamma} holds for the lifts, since
$d_H(\tilde\gamma, F(\tilde\gamma)) \geq d_H(\gamma, f(\gamma))$ holds - infact for all choice of lifts $F$ and $\tilde\gamma$.
  Moreover, since $F$ is injective, for each $i\in{\mathbb N}$ we have
$F^i(\tilde\gamma)\cap F^{i+1}(\tilde \gamma)=\emptyset$ and so from the order of boundary points we have for all $i\in{\mathbb N}$,
\begin{eqnarray}
		\tilde\gamma<F(\tilde \gamma)<F^2(\tilde\gamma)< \cdots < F^{i}(\tilde \gamma).
\end{eqnarray}
 Thus it suffices to show that
\begin{eqnarray}
\label{eq rightbound}
		F^{q({\alpha})-1}(\tilde \gamma)<T\tilde\gamma
\end{eqnarray}
and it will follow that the iterates $F(\tilde \gamma),\ldots, F^{q({\alpha})-1}(\tilde \gamma)$ all lie strictly in the region between $\tilde \gamma$ and $T(\tilde \gamma)$
in $\tilde{\mathbb A}$ and therefore that the iterates $f(\gamma),\ldots, \linebreak f^{q({\alpha})-1}(\gamma)$ are all disjoint from $\gamma$ as required.

To show (\ref{eq rightbound}) we argue by contradiction.   Assuming (\ref{eq rightbound}) is not true, we have
\begin{eqnarray}
\label{eq Qgood intersection}
		F^{q({\alpha})-1}(\tilde \gamma)\cap T\tilde\gamma\neq\emptyset
\end{eqnarray}
because of the order of boundary points. We will show that this means that $F^{q({\alpha})-1}(\tilde \gamma)$ and $F^{q({\alpha})}(\tilde \gamma)$ pass somewhere very
close to each other, because $F^{q({\alpha})}(\tilde \gamma)$ is close to $T\tilde\gamma$.   This will contradict \eqref{eq def c0}.

First, notice that by ${\alpha} \in \mathcal L_r(K_{r+1})$, \eqref{eq c''issmall} and that $\rho_*$ is decreasing, we have
\begin{eqnarray*}
\|\rho(f^{q({\alpha})})\|_{{\mathbb R}/{\mathbb Z}} &=& |q({\alpha}){\alpha}-1| ={\alpha}          \mathcal G({\alpha})<          \mathcal G({\alpha})<C''({\alpha},K_{r+1}^{-1})\\
&<& \rho_*(K_{r+1}^{{\alpha}^{-1}}) \leq \rho_*(\| Df\|^{q({\alpha})}) \leq \rho_*(\| Df^{q({\alpha})}\|).
\end{eqnarray*}
Then by \eqref{lem Fidfid}, we have
\begin{eqnarray*}
d_{H}( F^{q({\alpha})}(\tilde \gamma), T\tilde \gamma ) \leq   \| F^{q({\alpha})}- T\|  =   \| T^{-1}F^{q({\alpha})}- {\rm Id}\|  =   \| f^{q({\alpha})}- {\rm Id}\|.
\end{eqnarray*}
The last equality can be justified by Lemma \ref{lem:lift}, just as in the proof of Lemma \ref{lem no almost fixed point}.
  Thus by Theorem \ref{thm aflxz} and $K_{r+1} \geq K_1$, we have
\begin{eqnarray}
	d_{H}( F^{q({\alpha})}(\tilde \gamma), T\tilde \gamma ) &\leq&  A_0(\|\rho(f^{q({\alpha})})\|_{{\mathbb R}/{\mathbb Z}},K_1^{q({\alpha})}) \nonumber\\
											&\leq& A_0({\alpha}          \mathcal G({\alpha}), K_{r+1}^{q({\alpha})}).  \label{eq Qgood Hausdorff}
\end{eqnarray}
Then along with \eqref{eq def c0}, \eqref{eq def cL}, and the hypothesis ${\alpha} \in \mathcal L_r( K_{r+1})$, we have
\begin{eqnarray*}
d_H( F^{q({\alpha})-1}(\tilde \gamma), T\tilde \gamma ) &\geq& d_H( F^{q({\alpha})}(\tilde \gamma), F^{q({\alpha})-1}(\tilde \gamma) ) - d_H( F^{q({\alpha})}(\tilde \gamma), T\tilde \gamma ) \\
&\geq& K_{r+1}^{-q({\alpha})}C'_r({\alpha}^{-1}, K_{r+1})^{-1} - A_0({\alpha}          \mathcal G({\alpha}), K_{r+1}^{q({\alpha})}) > 0.
\end{eqnarray*}
However this contradicts \eqref{eq Qgood intersection}. Thus we have \eqref{eq rightbound}.
\begin{comment}
From (\ref{eq Qgood intersection}) and (\ref{eq Qgood Hausdorff}) we conclude that there exist points $z\in F^{q({\alpha})-1}(\tilde \gamma)$, $z'\in F^{q({\alpha})}(\tilde \gamma)$
satisfying
\begin{eqnarray}
\label{eq Qgood points}
									d(z,z')\leq A_0({\alpha}          \mathcal G({\alpha}), K_1^{q({\alpha})}).
\end{eqnarray}
On the other hand, the map $F^{1-q({\alpha})}:=(F^{q({\alpha})-1})^{-1}$ maps $z$ into $\tilde\gamma$ and maps $z'$ into $F(\tilde\gamma)$.   Therefore, composing with the inverse of the
admissible coordinates $\psi^{-1}$ these are mapped to points in $\{0\}\times[0,1]$ and $\{1\}\times[0,1]$ respectively, which are therefore at least distance $1$ apart.
Thus,
\begin{eqnarray}
		1\ &\leq&\  d\Big(\psi^{-1}\circ F^{1-q({\alpha})}(z),\  \psi^{-1}\circ F^{1-q({\alpha})}(z')\Big)\nonumber\\
							&\leq&\ \|D(\psi^{-1})\|\| D(F^{1-q({\alpha})})\|\cdot d(z,z')\nonumber\\
							&\leq&\ \|\psi\|_{\diff^1}\,\| f^{q({\alpha})-1}\|_{\diff^1}\,A_0({\alpha}          \mathcal G({\alpha}), K_1^{q({\alpha})})\nonumber
\end{eqnarray}
using (\ref{eq Qgood points}) in the last line.   Using Lemma \ref{lem brouwer curve compactness} for this value of $r$, and Theorem \ref{thm aflxz},
we obtain
\begin{eqnarray}
		1\ &\leq&\ C'_r(\rho(f)^{-1}, K_{r+1})\cdot K_1^{q({\alpha})-1}\cdot A_0({\alpha}          \mathcal G({\alpha}), K_1^{q({\alpha})}).\nonumber
\end{eqnarray}
By monotonicity we may replace each $K_1$ with $K_{r+1}$ in this inequality.  But this contradicts (\ref{eq def cL}) for $K=K_{r+1}$, completing the proof.

\end{comment}
\end{proof}

\section{Smooth domain bounded by good curves}

\begin{comment}

\subsection{Renormalization of pseudo-rotations}
In this section we explain how one iterate $f^k$ of a pseudo-rotation $f$ can in some sense be quotiented out by another iterate $f^l$ to yield a
new pseudo-rotation.   Then we will discuss how a Brouwer curve for the quotient map lifts to a Brouwer curve for higher iterates of
the original map $f^k$.

The `standard' annulus ${\mathbb A}$ can be canonically identified with the quotient $\tilde{\mathbb A}/T$ of the strip by the action of the translation
$T:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$, $(x,y)\mapsto(x+1,y)$.   More generally, if
$S:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$ is a
diffeomorphism that
acts freely and properly discontinuously, then the quotient space
\begin{eqnarray*}
					\tilde{\mathbb A}/S\ :=\ \Big\{\ [z]_S\ \ \Big|\ \ z\sim z' \ \iff\ \exists k\in{\mathbb Z}\ \mbox{ s.t. }\ S^kz=z'\ \Big\}
\end{eqnarray*}
is (non-canonically) diffeomorphic to the annulus ${\mathbb A}$, and the quotient map
\begin{eqnarray*}
			p_{S}: \tilde{\mathbb A} \rightarrow \tilde{\mathbb A}/S, \qquad z\mapsto [z]_S
\end{eqnarray*}
is a covering map whose deck transformation group is freely generated by $S$.    Each diffeomorphism $F:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$
descends to a diffeomorphism
\begin{eqnarray*}
					F_S :\tilde{\mathbb A}/S\rightarrow\tilde{\mathbb A}/S
\end{eqnarray*}
if and only if $FS=SF$.

In particular, if $f:{\mathbb A}\rightarrow{\mathbb A}$ is a pseudo-rotation and $S$ is a lift of $f^k$, some $k\neq0$, then, by Guillou's strong refinement of
Brouwer's plane translation theorem \cite[Th\'eor\`em 5.1]{L}, \red{$S$ acts freely and properly discontinuously\marginpar{\tiny stronger than\\ necessary: clean this up}} so that
the quotient space $\tilde{\mathbb A}/S$ is well defined, as described above.   Now, if $F$ is a lift of some other iterate $f^l$
then since $F$ and $S$ clearly commute, we see that $F$ descends to a map on the quotient space $F_S:\tilde{\mathbb A}/S\rightarrow\tilde{\mathbb A}/S$.
If $F$ and $S$ are not iterates of a common lift of $f$, then this quotient map can be non-trivial:

\begin{lemma}\label{lem renormalizing rot numbers}
	Let $f:{\mathbb A}\rightarrow {\mathbb A}$ be a pseudo-rotation.   Let $S:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$ be a lift of $f^k$, and
	$F$ be a lift of $f^l$, for some $k,l\in{\mathbb N}$.   If
	\begin{eqnarray}\label{irrational quotient}
								\rho(F)/\rho(S)\notin{\mathbb Q}
	\end{eqnarray}
	then the quotient map $F_S:\tilde{\mathbb A}/S\rightarrow\tilde{\mathbb A}/S$ is also a pseudo-rotation and its rotation number is given by
	\begin{eqnarray}\label{rotation number of quotient 0}
						\rho(F_S)=\rho(F)/\rho(S)\mod 1.
	\end{eqnarray}
\end{lemma}
\begin{proof}
Fix a lift $F_0$ of $f$ as a reference.   By this we mean a lift via the projection $p_T:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}/T$ after identifying ${\mathbb A}$ with $\tilde{\mathbb A}/T$,
so that $F_0$ commutes with $T$.   Since $S$ is a lift of $f^k$ and $F$ is a lift of $f^l$ we can write
\begin{eqnarray*}
				S=T^mF_0^k\qquad F=T^nF_0^l
\end{eqnarray*}
for some $m,n\in{\mathbb Z}$.   Condition (\ref{irrational quotient}) can be expressed as
\begin{eqnarray}\label{irrational quotient 1}
					\frac{k{\alpha} +m}{l{\alpha}+n}\notin{\mathbb Q}
\end{eqnarray}
where ${\alpha}=\rho(F_0)$.   In particular, at least one of $m$ and $n$ must be non-zero.
Now to show that $F_S$ has no periodic orbits we argue indirectly.   Suppose that
\begin{eqnarray*}
			F_S^a([z]_S)=[z]_S
\end{eqnarray*}
for some $[z]_S\in\tilde{\mathbb A}/S$ and $a\in{\mathbb Z}\backslash\{0\}$.   Then
\begin{eqnarray*}
			 F^a(z)=S^j(z)
\end{eqnarray*}
for some $j\in{\mathbb Z}$.   Thus $(T^nF_0^l)^a(z)=(T^mF_0^k)^j(z)$ and so
\begin{eqnarray}\label{proof renormalizing 2}
			 F_0^{kj-al}(z)=T^{an-jm}(z).
\end{eqnarray}
Applying the projection $p_T:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}/T$ gives
\begin{eqnarray*}
				f^{kj-al}([z]_T)=[z]_T
\end{eqnarray*}
so that $f$ has a $(kj-al)$-periodic point.
To get a contradiction it remains to show that $kj-al\neq0$.   But if $kj-al=0$ then (\ref{proof renormalizing 2}) implies $T^{an-jm}(z)=z$, which implies
$an-jm=0$, which implies $j(k{\alpha} +m)/(l{\alpha}+n)a=1$ contradicting (\ref{irrational quotient 1}).   We conclude that $F_S$ has no periodic points.

To show (\ref{rotation number of quotient 0}), it suffices to compute the rotation number of the restriction of $F_S$ to a circle map on a boundary
component.   This follows from the following general observation.   Consider a pair of orientation preserving homeomorphisms
$S,F:{\mathbb R}\rightarrow{\mathbb R}$ that arise as lifts of circle homeomorphisms on ${\mathbb R}/{\mathbb Z}$, and suppose that $S$ has no fixed points so that it acts freely
and properly discontinuously on ${\mathbb R}$.   In particular the quotient ${\mathbb R}/S$ is a circle.   If $F$ and $S$ commute then $F$ descends to a map
$F_S:{\mathbb R}/S\rightarrow {\mathbb R}/S$ and it is not hard to verify that its rotation number is given by
\begin{eqnarray*}
				\rho(F_S)=\rho(F)/\rho(S)\mod 1.
\end{eqnarray*}
\end{proof}

\begin{comment}

\begin{lemma}\label{lem: quotients 1}
Suppose $S:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$ is a bijection without periodic points.
Let $\sigma\subset \tilde{\mathbb A}/S$ be the projection under $p_{S}: \tilde{\mathbb A} \rightarrow \tilde{\mathbb A}/S$ of a map $\tilde\sigma:I\subset{\mathbb R}\rightarrow\tilde{\mathbb A}$.
Then $\sigma$ is simple if and only if $\tilde\sigma$ is simple and
\begin{eqnarray}\label{quotients simple}
	\qquad\tilde\sigma\cap S^k\tilde\sigma=\emptyset\qquad \forall k\in{\mathbb Z}\backslash\{0\}.
\end{eqnarray}
Suppose moreover that $F:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$ is a map that commutes with $S$ (so that $F_S$ is well defined).
Then $\sigma$ is a Brouwer curve for $F_S$, in the sense that $F_S(\sigma)\cap \sigma=\emptyset$, if and only if
\begin{eqnarray}\label{quotients Brouwer}
\qquad F\tilde\sigma\cap S^k\tilde\sigma=\emptyset\qquad \forall k\in{\mathbb Z}.
\end{eqnarray}
\end{lemma}
\begin{proof}
This follows easily from the definitions.   \marginpar{\tiny Proof just for B to be \\sure.\\  Final version can leave proof for reader.}

Consider first the conditions for simplicity.    Suppose $\sigma$ is simple.   Then clearly $\tilde\sigma$ is simple,
since $\tilde\sigma(t_0)=\tilde\sigma(t_1)$ for some $t_0\neq t_1$ implies $\sigma(t_0)=\sigma(t_1)$.    To show (\ref{quotients simple}) holds, suppose
for a contradiction that $\tilde\sigma\cap S^k\tilde\sigma\neq\emptyset$ for some $k\in{\mathbb Z}\backslash\{0\}$.    Then we find $t_0,t_1$ with
\begin{eqnarray}\label{proof: quotients 1}
	\tilde\sigma(t_0)= S^k\tilde\sigma(t_1).
\end{eqnarray}
As $S^k$ has no fixed points this means $\tilde\sigma(t_0)\neq \tilde\sigma(t_1)$ so that $t_0\neq t_1$.   But then projecting (\ref{proof: quotients 1})
gives $\sigma(t_0)=\sigma(t_1)$ contradicting simplicity of $\sigma$.    Conversely, suppose that $\tilde\sigma$ is simple and that (\ref{quotients simple}) holds.
Then $\sigma$ is also simple, else we find $t_0\neq t_1$ with $\sigma(t_0)=\sigma(t_1)$, which means that $\tilde\sigma(t_0)$ and $\tilde\sigma(t_1)$ project to
the same point, which means there exists $k\in{\mathbb Z}$ so that $\tilde\sigma(t_0)= S^k\tilde\sigma(t_1)$.   As $\tilde\sigma$ is simple we have $k\neq0$.   But
then this contradicts (\ref{quotients simple}).

Now consider the conditions for $\sigma$ to be a Brouwer curve.   If $F_S(\sigma)\cap \sigma=\emptyset$ then for all $k\in{\mathbb Z}$ we have
$p_S(F(\tilde\sigma)\cap S^k\tilde\sigma)\subset p_S\circ F(\tilde\sigma)\cap p_S\circ S^k(\tilde\sigma)=F_S(\sigma)\cap \sigma=\emptyset$ giving
(\ref{quotients Brouwer}).    Conversely, suppose that (\ref{quotients Brouwer}) holds.   Then we claim that $F_S(\sigma)\cap \sigma=\emptyset$.
Indeed, if not, then we find points $z\in F(\tilde\sigma)$ and $z'\in\tilde\sigma$ which project to the same point (in $F_S(\sigma)\cap \sigma$).
Thus there exists $k\in{\mathbb Z}$ so that $z=S^kz'$.   But then $z\in F(\tilde\sigma)\cap S^k\tilde\sigma$ contradicting (\ref{quotients Brouwer}).
\end{proof}

\begin{prop}\label{prop smooth return domain}
Let $f:{\mathbb A}\rightarrow {\mathbb A}$ be a pseudo-rotation.  Set ${\alpha}:=\rho(f)\in(0,1)$.
For each $n\in{\mathbb N}$ let $F_n$ be the unique lift of $f^{q_n}$ having $\rho(F_n)\in(-1/2,1/2)$.
Suppose that $\gamma\subset\tilde{\mathbb A}/F_n$ is an $a_{n+2}$-good Brouwer curve for the quotient map
\begin{eqnarray*}
			{(F_{n+1})}_{F_{n}}:\tilde{\mathbb A}/F_n\rightarrow \tilde{\mathbb A}/F_n.
\end{eqnarray*}
Let $\tilde\gamma\subset\tilde{\mathbb A}$ be any lift of $\gamma$.   Then the projection
\begin{eqnarray*}
	\hat\gamma:=p_T(\tilde\gamma)\subset \tilde{\mathbb A}/T
\end{eqnarray*}
is a $q_{n+2}$-good Brouwer curve for $f$, where $q_0,q_1,\ldots$ and $a_1,a_2,\ldots$ are the sequences in ${\mathbb N}$ associated to ${\alpha}$.
\end{prop}
\begin{proof}
First note that for each $n\in{\mathbb N}$ we have $\rho(F_n)=q_n{\alpha}-p_n$, so that
\begin{eqnarray}\label{rotation numbers}
				\mathrm{sign}\,\rho(F_n)=(-1)^n	,
\end{eqnarray}
see (\ref{sign}), and from uniqueness considerations,
\begin{eqnarray}\label{representation F_n}
						F_{n}=T^{-p_n}F^{q_n}
\end{eqnarray}
where $F:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$ is the unique lift of $f$ with $\rho(F)={\alpha}$.

Fix $n\in{\mathbb N}$.   Applying Lemma \ref{lem: quotients 1} we can reformulate the assumptions and conclusions of the Proposition in terms of $\tilde\gamma$.   Indeed,
as a special case of (\ref{quotients simple}), $\tilde\gamma$ is a simple regular curve satisfying
\begin{eqnarray}\label{quotients assumption simple}
			F_n(\tilde\gamma)\cap \tilde\gamma=\emptyset
\end{eqnarray}
because the projection $p_{F_n}(\tilde\gamma)\subset\tilde{\mathbb A}/F_n$ is a simple regular curve.
By (\ref{quotients Brouwer})
\begin{eqnarray}\label{quotients assumption Brouwer}
			F_{n+1}^{i}(\tilde\gamma)\cap F_{n}^{j}(\tilde\gamma)=\emptyset \qquad\forall l\in{\mathbb Z}
\end{eqnarray}
holds for each $i\in\{1,\ldots,a_{n+2}-1\}$, because the projection $p_{F_n}(\tilde\gamma)$ is an $a_{n+2}$-good Brouwer curve for the quotient map ${(F_{n+1})}_{F_{n}}$.
To show $\hat\gamma$ is a simple regular curve, we must show that $\tilde\gamma$ is simple (which we have just observed is the case), and that
\begin{eqnarray}\label{quotients conclusion simple}
			T^l(\tilde\gamma)\cap \tilde\gamma=\emptyset \qquad\forall l\in{\mathbb Z}\backslash\{0\},
\end{eqnarray}
while to show that $\hat\gamma$ is a $q_{n+2}$-good Brouwer curve for $f$, we must show
\begin{eqnarray}\label{quotients conclusion Brouwer}
			F^k(\tilde\gamma)\cap T^l\tilde\gamma=\emptyset \qquad\forall l\in{\mathbb Z}
\end{eqnarray}
for all $k\in\{1,\ldots,q_{n+2}-1\}$.    In short, we must show that (\ref{quotients assumption simple}) and (\ref{quotients assumption Brouwer}) imply
(\ref{quotients conclusion simple}) and (\ref{quotients conclusion Brouwer}).

First let us show (\ref{quotients conclusion simple}).   By applying iterates of $T$, and using that for two simple regular curves $\gamma_1,\gamma_2$
the relation $\gamma_1<\gamma_2$ is equivalent to $T\gamma_1<T\gamma_2$, it suffices to prove (\ref{quotients conclusion simple}) for $l=1$ or $l=-1$.
Now we note that
\begin{eqnarray}\label{T from quotient maps}
							(F_{n})^{q_{n+1}}(F_{n+1}^{-1})^{q_{n}}=T^{(-1)^n}
\end{eqnarray}
which follows from (\ref{representation F_n}) and $p_{n+1}q_{n}-p_{n}q_{n+1}=(-1)^n$, see (\ref{relation pn qn}).
By (\ref{quotients assumption simple}) and (\ref{quotients assumption Brouwer}) with $i=1$ and $j=0$, we know that $F_{n}$ and $F_{n+1}$ each map
$\tilde\gamma$ to a curve disjoint from $\tilde\gamma$.   Moreover from (\ref{rotation numbers}) we see that $F_{n}$ and $(F_{n+1})^{-1}$ both
map $\tilde\gamma$ strictly to the same side of $\tilde\gamma$ (both to the right of $\tilde\gamma$ when $n$ is even, else both to the left of $\tilde\gamma$).
Thus by (\ref{T from quotient maps}) we see that $T^{(-1)^n}$ maps $\tilde\gamma$ strictly to the right or left of $\tilde\gamma$.   In either case
(\ref{quotients conclusion simple}) holds for $l=(-1)^{n}$ as required.

It remains to show (\ref{quotients conclusion Brouwer}).    Note the relation
\begin{eqnarray}\label{F from quotient maps}
							(F_{n})^{p_{n+1}}(F_{n+1}^{-1})^{p_{n}}=F^{(-1)^n}
\end{eqnarray}
which again follows from (\ref{representation F_n}) and (\ref{relation pn qn}).
Substituting (\ref{F from quotient maps}) and (\ref{T from quotient maps}) into (\ref{quotients conclusion Brouwer}),
we see that (\ref{quotients conclusion Brouwer}) is equivalent to
\begin{eqnarray*}
		\qquad \big[(F_{n})^{p_{n+1}}(F_{n+1}^{-1})^{p_{n}}\big]^{{(-1)^n}k}(\tilde\gamma)\ \cap\ \big[(F_{n})^{q_{n+1}}(F_{n+1}^{-1})^{q_{n}}\big]^{{(-1)^n}l}(\tilde\gamma)   = \emptyset
\end{eqnarray*}
which is equivalent to
\begin{eqnarray}\label{T from quotient maps 1}
		     (F_{n})^{{(-1)^n}(kp_{n+1}-lq_{n+1})}\circ (F_{n+1}^{-1})^{{(-1)^n}(kp_n-lq_n)}(\tilde\gamma)\ \cap\ \tilde\gamma   = \emptyset.
\end{eqnarray}
Since $F_{n}, F_{n+1}, T$ are all orientation preserving homeomorphisms of $\tilde{\mathbb A}$ that preserve the two boundary components, one knows that for any
two simple regular curves $\gamma_1,\gamma_2$ the relation $\gamma_1<\gamma_2$ is preserved under any of these maps and their inverses.
So using again that $F_{n}$ and $(F_{n+1})^{-1}$ both map $\tilde\gamma$ strictly to the same side of itself, it follows that a sufficient condition for
(\ref{T from quotient maps 1}) to hold, is that the two exponents
\begin{eqnarray*}
			\epsilon_{n+1}(k,l):= kp_{n+1}-lq_{n+1}\qquad \epsilon_{n}(k,l):= kp_{n}-lq_{n}
\end{eqnarray*}
have the same sign.   More precisely, abbrieviating $\epsilon_{j}(k,l)$ by $\epsilon_{j}$,
it is enough to show that either both $\epsilon_{n}$ and $\epsilon_{n+1}$
are non-vanishing and have the same sign, or that one vanishes and the other is non-vanishing.   Let us now restrict to $k\geq1$.
Then $\epsilon_{j}=0$ is equivalent to $p_j/q_j=l/k$, so it follows from $p_n/q_n\neq p_{n+1}/q_{n+1}$ that $\epsilon_{n+1}$ and $\epsilon_{n}$ cannot
both vanish.   Therefore a sufficient condition for (\ref{quotients conclusion Brouwer}) to hold for $(k,l)\in{\mathbb N}\times{\mathbb Z}$ is
\begin{eqnarray*}
				\epsilon_{n}\epsilon_{n+1}\geq 0.
\end{eqnarray*}
So to complete the proof of the Proposition it is enough to show
\begin{eqnarray}\label{sign of en}
	\epsilon_{n}\epsilon_{n+1}< 0\ \implies\ k\geq q_{n+2}.
\end{eqnarray}
We note that $\epsilon_j<0$ if and only if $p_j/q_j<l/k$.   For $n$ even one has
\begin{eqnarray*}
				\frac{p_n}{q_n} < \frac{p_{n+1}}{q_{n+1}}
\end{eqnarray*}
and the inequality is reversed for $n$ odd.   So for $n$ even $\epsilon_{n}\epsilon_{n+1}< 0$ implies $\epsilon_{n}<0<\epsilon_{n+1}$, which implies
\begin{eqnarray}\label{interval n even}
				\frac{l}{k}\in \left(\frac{p_n}{q_n},\ \frac{p_{n+1}}{q_{n+1}}\right).
\end{eqnarray}
Similarly for $n$ odd $\epsilon_{n}\epsilon_{n+1}< 0$ implies $\epsilon_{n+1}<0<\epsilon_{n}$ which implies
\begin{eqnarray}\label{interval n odd}
				\frac{l}{k}\in \left(\frac{p_{n+1}}{q_{n+1}},\ \frac{p_{n}}{q_{n}}\right).
\end{eqnarray}
In both cases the given interval also contains $\alpha$, and for each $j\geq1$, $p_j/q_j$ is the best rational approximation to $\alpha$ amongst fractions with denominator less than or equal to $q_j$.    Thus (\ref{interval n even}) or (\ref{interval n odd}) imply that $k> q_n$
\footnote{This completes the proof of (\ref{quotients conclusion Brouwer}) for all
$k\in\{1,\ldots,q_n\}$, and therefore that $\hat\gamma$ is a $q_{n}$-good Brouwer curve for $f$.   Note that up to here we have only used (\ref{quotients assumption Brouwer})
for $i=1$, i.e.\ that $\gamma$ is a Brouwer curve (rather than a $q_{n+2}$-good Brouwer curve) for $(F_{n+1})_{F_n}$.}.
It remains to show that $k\geq q_{n+2}$.

From the theory of rational approximation, when either (\ref{interval n even}) or
(\ref{interval n odd}) hold, $k$ and $l$ must have the special form
\begin{eqnarray}\label{special pairs}
			(l,k)\in\big\{\ (p_n + ip_{n+2}, q_n + iq_{n+2})\ |\ 1\leq i< a_{n+2}\ \big\}.
\end{eqnarray}
Thus it remains only to show (\ref{quotients conclusion Brouwer}) for pairs $(l,k)$ of the form in (\ref{special pairs}).
These cases are ruled out easily from (\ref{quotients assumption Brouwer}).    Indeed, setting $j=-1$ in (\ref{quotients assumption Brouwer}) gives
$F_{n+1}^{i}(\tilde\gamma)\cap F_{n}^{-1}(\tilde\gamma)=\emptyset $
for each $i\in\{1,\ldots,a_{n+2}-1\}$.   Equivalently,
\begin{eqnarray*}
			\big(T^{-p_{n+1}}F^{q_{n+1}}\big)^{i}(\tilde\gamma)\cap \big(T^{-p_{n}}F^{q_{n}}\big)^{-1}(\tilde\gamma)=\emptyset
\end{eqnarray*}
hence
\begin{eqnarray*}
			F^{q_n+iq_{n+1}}(\tilde\gamma)\cap T^{p_{n}+ip_{n+1}}(\tilde\gamma)=\emptyset
\end{eqnarray*}
which is (\ref{quotients conclusion Brouwer}) for $k,l$ of the form in (\ref{special pairs}).   This completes the proof.
\end{proof}

To apply the estimates from section 2 to a quotient map we first need to identify the quotient space on which it acts with the standard annulus ${\mathbb A}\simeq \tilde{\mathbb A}/T$ in some way,
so that we can make sense of $C^r$-bounds.   This follows immediately from Lemma \ref{lem brouwer curve compactness} applied to the quotient map $(F_{n+1})_{F_n}$:

\begin{lemma}\label{quotient representation}
Let $f:{\mathbb A}\rightarrow {\mathbb A}$ be a $\{K_r\}_{r\geq1}$-smooth $\omega$-preserving pseudo-rotation and set ${\alpha}:=\rho(f)\in(0,1)$.
Let $F$ be the unique lift of $f$ having $\rho(F)={\alpha}$.   If ${\alpha}\in\mathcal L(K_{1})$, then for each $r\geq 1$ there exists a smooth diffeomorphism
\begin{eqnarray*}
		\psi=\psi_r:\tilde{\mathbb A}/T\rightarrow\tilde{\mathbb A}/F
\end{eqnarray*}
with a lift $\tilde\psi:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$ that pulls back $\omega_0$ to a constant multiple of $\omega_0$,
and which satisfies
\begin{eqnarray}\label{representation estimate}
			\|\tilde\psi\|_{\mathrm{Diff}^r([0,1]^2)}\leq C'_r({\alpha}^{-1},K_{r+1}).
\end{eqnarray}
\end{lemma}
\begin{proof}
By Lemma \ref{lem brouwer curve compactness},
$f$ has a Brouwer curve $\gamma=\gamma_r\subset{\mathbb A}/T$ for which the region
$\Omega\subset{\mathbb A}/T$ bounded by $(\gamma,f(\gamma))$ admits an $\big(r,C'_r({\alpha}^{-1},K_{r+1})\big)$-admissible coordinate
\begin{eqnarray*}
				\psi:[0,1]^2\rightarrow \Omega
\end{eqnarray*}
for $f$.
Let $\tilde\gamma$ be a lift of $\gamma$ and $\tilde\Omega\subset\tilde{\mathbb A}$ be the region bounded by $(\tilde\gamma,F(\tilde\gamma))$.
The projection $p_T:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}/T$ is a diffeomorphism from $\tilde\Omega$ to $\Omega$ whose differential is the identity everywhere,
so that composing $\psi$ with the inverse of $p_T$ gives us a map
$\tilde\psi:[0,1]^2\rightarrow\tilde\Omega$ which also satisfies the estimate (\ref{representation estimate}), and which, from the definition of admissible coordinate, has constant Jacobian
and extends uniquely to a continuous map $\tilde\psi:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$ (denoted with the same symbol), that satisfies
\begin{eqnarray*}
						\tilde\psi\circ T = F\circ\tilde\psi.
\end{eqnarray*}
Thus $\tilde\psi$ is the lift of a diffeomorphism $\tilde{\mathbb A}/T\rightarrow\tilde{\mathbb A}/F$ satisfying the desired conditions.
\end{proof}

Now we can prove the main result of this section.

\begin{prop} \marginpar{\tiny maybe need version for $n$ even instead}
For each integer  $r \geq 1$, there exist increasing functions $P_r:  {\mathbb R}^2_{+} \to {\mathbb R}_+$ and  $ W_r : {\mathbb N} \times  {\mathbb R}_+ \to {\mathbb R}_+$  such that  the following is true.
Suppose $f:{\mathbb A}\rightarrow {\mathbb A}$ is a $\{K_r\}_{r\geq1}$-smooth $\omega$-preserving pseudo-rotation with $\rho(f) = {\alpha}$.    If \textbf{(condition on ${\alpha}$ in terms of $P_r$)}, then for $n$ odd,
$f$ has a $q_{n+2}$-good Brouwer curve $\gamma$ such that the region in ${\mathbb A}$ bounded by $(\gamma, f^{q_{n+1}}(\gamma))$ admits a
$(r, W_r( q_{n+2}, K_{r+3}))$-admissible coordinate for $f^{q_{n+1}}$.
\end{prop}
\begin{proof}
We wish to apply Lemma \ref{quotient representation} to the pseudo-rotation $f^{q_n}:\tilde{\mathbb A}/T\rightarrow \tilde{\mathbb A}/T$ to obtain a
change of coordinates
\begin{eqnarray}\label{coordinate change}
		\psi:\tilde{\mathbb A}/T\rightarrow\tilde{\mathbb A}/F_n.
\end{eqnarray}
To do this we observe that since $f:{\mathbb A}\rightarrow {\mathbb A}$ is a $\{K_r\}_{r\geq1}$-smooth pseudo-rotation with $\rho(f) = {\alpha}$, so is $f^{q_n}$ an
$\{M_r\}_{r\geq 1}$-smooth pseudo-rotation with
\begin{eqnarray*}
					\rho(f^{q_n})=q_n{\alpha} - p_n = (-1)^n\beta_n,
\end{eqnarray*}
(see \ref{sign}), where each $M_r$ depends only on $K_r$ and $q_n$ and not for example on $f$.    Now if
\begin{eqnarray*}
						\beta_n\in\mathcal L(M_1),
\end{eqnarray*}
then indeed by Lemma \ref{quotient representation}, for each $r\geq 1$ such a diffeomorphism $\psi=\psi_r$ as in (\ref{coordinate change}) exists, and has a
lift $\tilde\psi:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$ that pulls back $\omega_0$ to a constant multiple of $\omega_0$,
and
\begin{eqnarray}\label{coordinate change estimate}
			\|\tilde\psi\|_{\mathrm{Diff}^r([0,1]^2)}\leq C'_r(\beta_n^{-1},M_{r+1}).
\end{eqnarray}
Let us assume this $r\geq1$ is now fixed.
Let $g$ uniquely complete the diagram
\begin{eqnarray}\label{diagram}
	\xymatrix{
			\tilde{\mathbb A}/F_n \ar[r]^{(F_{n+1})_{F_n}} & \tilde{\mathbb A}/F_n  \\
			\tilde{\mathbb A}/T \ar[u]^{\psi}\ar[r]_{g} & \tilde{\mathbb A}/T \ar[u]_{\psi}
	}
\end{eqnarray}
So $g$ is nothing other than the quotient map $(F_{n+1})_{F_n}$ in the coordinates of $\psi$, and has the advantage that we can talk of $\mathrm{Diff}^r$-bounds for $g$ and
in particular can apply Lemma \ref{lem brouwer curve compactness}.

What are the $\mathrm{Diff}^r$-bounds for $g$?    The map $G:=\tilde{\psi}^{-1}\circ F_{n+1}\circ\tilde\psi$ is a lift of $g$, and
$\|G\|_{\mathrm{Diff}^r([0,1]^2)}\leq D_0\left(\|\tilde{\psi}\|_{\mathrm{Diff}^r([0,1]^2)},\,\|F_{n+1}\|_{\mathrm{Diff}^r(\tilde{\psi}([0,1]^2))}\right)$
for some continuous increasing function $D_0:[0,\infty)^2\rightarrow[0,\infty)$.   Infact $\tilde{\psi}([0,1]^2))\subset[-T,T]\times[0,1]$ for some $T>0$
depending only on $\|\tilde\psi\|_{\mathrm{Diff}^1([0,1]^2)}$ and therefore by (\ref{coordinate change estimate}) only on $\beta_n$ and $M_{2}$, which
in turn depends only on $\beta_n$, $q_n$ and $K_2$.   Thus $\|F_{n+1}\|_{\mathrm{Diff}^r(\tilde{\psi}([0,1]^2))}$ is bounded by a constant depending
only on $K_r, q_{n+1}, q_n, \beta_n, K_2$.   So for $r\geq2$, \marginpar{\tiny hence $\leq D_1(K_{r+1},q_{n+1}, q_n, \beta_n)$}
\arys
			\qquad \|G\|_{\mathrm{Diff}^r([0,1]^2)}\leq D_r=D\left(C'_r(\beta_n^{-1},M_{r+1}),K_r,q_{n+1}, q_n, \beta_n\right)
\end{eqnarray*}
some sequence $\{D_r\}_{r\geq1}$ depending on
\begin{eqnarray}\label{bounds D_r}
				C'_r(\beta_n^{-1},\ M_{r+1}),\ K_r,\ q_{n+1},\ q_n,\ \beta_n.
\end{eqnarray}
This expresses our $\mathrm{Diff}^r$-bounds for $g$.   The rotation number of $g$ is that of $(F_{n+1})_{F_n}$, and so
by Lemma \ref{lem renormalizing rot numbers},
\begin{eqnarray*}
			\rho(g)=\frac{\rho(F_{n+1})}{\rho(F_n)}=\frac{q_{n+1}{\alpha}-p_{n+1}}{q_{n}{\alpha}-p_{n}}=-\frac{\beta_{n+1}}{\beta_{n}}=-{\mathcal G}^{n+1}({\alpha}).
\end{eqnarray*}
Therefore, if
\begin{eqnarray*}
						-{\mathcal G}^{n+1}({\alpha})\in \mathcal L(D_1)
\end{eqnarray*}
then by Lemma \ref{lem brouwer curve compactness}, for each $r\in{\mathbb N}$ there exists a Brouwer curve $\gamma=\gamma_r$ for $g$ such that the region
$\Omega_0\subset\tilde{\mathbb A}/T$ bounded by $\big(\gamma,g(\gamma)\big)$
admits an $\big(r,\,C'_r({\mathcal G}^{n+1}({\alpha})^{-1}, D_{r+1})\big)$-admissible coordinate.   Pushing forward $\gamma$ via $\psi$,
see (\ref{diagram}), we see that
\begin{eqnarray*}
		\eta:=\psi(\gamma)\subset\tilde{\mathbb A}/F_n
\end{eqnarray*}
is a Brouwer curve for $(F_{n+1})_{F_n}$.    Applying Proposition \ref{prop smooth return domain} to $\eta$ we conclude that if
$\tilde\eta\subset\tilde{\mathbb A}$ is any lift of $\eta$ via the projection map $p_{F_n}:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}/F_n$, then
\begin{eqnarray*}
								\hat\eta:=p_T(\tilde\eta)\subset\tilde{\mathbb A}/T
\end{eqnarray*}
is a $q_{n+2}$-good Brouwer curve for $f$.

It remains to give bounds on admissible coordinates for the region $\Omega\subset\tilde{\mathbb A}/T$ bounded by
\begin{eqnarray*}
						\big(\hat\eta,f^{q_n}(\hat\eta)\big).
\end{eqnarray*}
Let $\tilde\gamma$ be a lift of $\gamma$ via $p_T:\tilde{\mathbb A}\rightarrow{\mathbb A}/T$.    Then the region $\tilde\Omega_0\subset\tilde{\mathbb A}$ bounded by
$\big(\tilde\gamma,G(\tilde\gamma)\big)$ pushes forward via $\tilde\psi$ to the region $\tilde\Omega\subset\tilde{\mathbb A}$ bounded by
\begin{eqnarray*}
					\big(\tilde\eta,F_{n+1}(\tilde\eta)\big)
\end{eqnarray*}
where $\tilde\eta:=\tilde\psi(\tilde\gamma)\subset\tilde{\mathbb A}$ is a lift of $\eta$.   The admissible coordinates for $\Omega_0$ lift to
$\big(r,\,C'_r({\mathcal G}^{n+1}({\alpha})^{-1}, D_{r+1})\big)$-admissible coordinates for $\tilde\Omega_0$ because $T$ is locally an isometry.
Pushing these coordinates forward by $\tilde\psi$ we see that the region $\tilde\Omega$ admits $(r,\,\tilde W_r)$-admissible coordinates,
where $\tilde W_r$ depends on $\|\tilde\psi\|_{\mathrm{Diff}^r(\tilde\Omega_0)}$ and $C'_r({\mathcal G}^{n+1}({\alpha})^{-1}, D_{r+1})$.
From (\ref{coordinate change estimate}) and (\ref{bounds D_r}) this means that $\tilde W_r$ depends on
\marginpar{\tiny are these controlled by $q_{n+2}, K_{r+3}$ ?}
\begin{eqnarray*}
						{\mathcal G}^{n+1}({\alpha})^{-1},\ C'_r(\beta_n^{-1},\ M_{r+1}),\ K_r,\ q_{n+1},\ q_n,\ \beta_n.
\end{eqnarray*}
Projecting $\tilde W_r$ via $p_T:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}/T$ gives us the same kind of admissible coordinates $W_r$ for the region $\Omega$
for the map $f^{q_{n+1}}$.  \\
\end{proof}

   \color{black}

\end{comment}

\subsection{Renormalization of pseudo-rotations}\label{Construction of a new pseudo-rotation}\label{S:gluing}
 Given a $C^\infty$ pseudo-rotation $f$ on ${\mathbb A}$ and an integer $n \geq 1$, we denote by  $F_n$ \textit{the unique lift of $f^n$ to $\tilde {\mathbb A}$ such that $\rho(F_n) \in (0, 1)$}.
 Given a smooth Brouwer curve $\gamma_n$ for $f^n$, we let $\tilde \gamma_n$ be an arbitrary lift of $\gamma_n$ to $\tilde {\mathbb A}$. We let  $\Omega_n$ be the unique closed region in $\tilde {\mathbb A}$ bounded by $(\tilde \gamma_n,F_n(\tilde \gamma_n))$.

 Given a $C^\infty$ admissible coordinate $
 H : [0,1]^2 \to  \Omega_n$ for $(\Omega_n,  F_n)$ (see Remark \ref{rema unique continuation}),
we can uniquely extend $H$ to a  $C^\infty$ diffeomorphism of $\tilde {\mathbb A}$ with constant Jacobian, denoted again by $H$, satisfying
\begin{eqnarray}\label{eq H}
HT = F_nH.
\end{eqnarray}
We notice that although the $C^r$ norms of $F_n$ and $T$ are uniformly bounded throughout $\tilde {\mathbb A}$, the $C^r$ norm of $H$ need not be uniformly bounded. However it is clear from \eqref{eq H} that for any integer $L > 0$, the norm $\|D^r H \|_{[- L, L] \times [0,1]}$ is bounded in terms of $L$ and $\|F_n\|_{\diff^r(\tilde {\mathbb A})}$.

Denote $ \alpha = \rho(f)  \in (0,1) \setminus {\mathbb Q}$. We have $\rho(F_1) = {\alpha}$.
 We abbrieviate
 \begin{eqnarray*}
 			F^{a,b} := T^{b}F_1^a \quad \forall a,b \in {\mathbb Z}.
\end{eqnarray*}
By our previous definition, we have $F_n = F^{n, -\lfloor n{\alpha} \rfloor}$.
Denote by $J: \tilde {\mathbb A} \to \tilde {\mathbb A}$ the orientation reversing diffeomorphism
\begin{eqnarray*}
J(x, y) = (-x,y).
\end{eqnarray*}
Notice that by \eqref{eq H} we have
\begin{eqnarray} \label{eq jtjtinverse}
 JH^{-1}F_n HJ = J T J = T^{-1}.
\end{eqnarray}
We set
\begin{eqnarray} \label{eq fabh}
F^{a,b}_H := JH^{-1}F^{a,b}HJ \quad \forall a, b \in {\mathbb Z}.
\end{eqnarray}
Since $F^{a,b}$ is $\omega$-preserving and commutes with $F_n$, and since $H$ has constant Jacobian, we deduce from \eqref{eq jtjtinverse} that $F^{a,b}_H$ is also  $\omega$-preserving, and commutes with $T$.
Consequently $F^{a,b}_H$ descends to an  $\omega$-preserving $C^\infty$ diffeomorphism $f^{a,b}_H : {\mathbb A} \to {\mathbb A}$.

We have the following lemma.

\begin{lemma}\label{prop. return map}
	There is a sequence of  increasing functions $\{ E_r : {\mathbb N}^2 \times {\mathbb R}_+^{ 2} \to {\mathbb R}_+  \}_{r \geq 1}$
	such that the following is true.
	Let $f$ be a $\{K_r\}_{r \geq 1}$-smooth  $\omega$-preserving pseudo-rotation with $\rho(f) = {\alpha}$; let $n \neq  0$ be an integer; let $\Omega_n$ be the region bounded by $(\tilde\gamma_n, F_n(\tilde\gamma_n))$ where $\tilde\gamma_n$ is some lift of a Brouwer curve for $f^n$. Assume that $(\Omega_n, F_n)$ admits  a $(r,L_r)$-admissible coordinate $H$.
	Then for any $a, b \in {\mathbb Z}$ with $a \lfloor n{\alpha} \rfloor + b n \neq 0$,
	$f^{a,b}_H$ is a  pseudo-rotation in $\diff^\infty({\mathbb A}, \omega)$  with
	\begin{eqnarray*}
	\rho(f^{a,b}_H) = \left\{ \frac{ -  a {\alpha} - b }{ \{ n{\alpha} \}} \right\}.
	\end{eqnarray*}   Moreover, we have
\begin{eqnarray*}
\norm{f^{a,b}_H}_{\diff^r} \leq E_r(|a|,|b|, K_r, L_r).
\end{eqnarray*}
\end{lemma}

\begin{proof}

	We first show that $f^{a,b}_H$ has no periodic points.
	Assume to the contrary there are integers $p \in {\mathbb Z}$, $q > 0$ and some $z \in \tilde{\mathbb A}$
    such that
    \begin{eqnarray*}
    (F^{a,b}_H)^q(z) = T^p(z).
    \end{eqnarray*}
    Then by \eqref{eq jtjtinverse} and \eqref{eq fabh} we have
	\begin{eqnarray} \label{eq periodicpoint}
	z  = (F^{a,b}_H)^q T^{-p}(z) = J H^{-1} F_1^{(qa + pn)} T^{ (qb - p\lfloor n{\alpha} \rfloor ) }  H J(z).
	\end{eqnarray}
	However, \eqref{eq periodicpoint} and the condition $a \lfloor n{\alpha} \rfloor + bn \neq 0$ implies that $HJ(z)$ descends to a perodic point for $f$, which contradicts the hypothesis that $f$ is a pseudo-rotation.
	We conclude that $f^{a,b}_H$ is a pseudo-rotation in $\diff^\infty({\mathbb A}, \omega)$.

	To compute $\rho(f^{a,b}_H)$, it suffices to study the trajectory of an arbitrary $z \in \tilde{\mathbb A}$ under the iterates of $(F^{a,b}_H)^q T^{-p}$ using the second equality in \eqref{eq periodicpoint}; for example a point on the boundary. This standard argument is left to the readers.

	For the $C^r$-norm of $f^{a,b}_H$, it suffices to control the $C^r$-norm of $F^{a,b}_H$ restricted to $(-1,2) \times [0,1]$.
	By \eqref{eq fabh}, $\| D^r F^{a,b}_H\|_{(-1,2) \times [0,1]}$ depends only on the $C^r$-norms of $F^{a,b}$, and the $C^r$-norm of $J$ and $H$ on
	\begin{eqnarray*}
	F^{a,b}_H((-1,2) \times [0,1]) \cup (-1,2) \times [0,1].
	\end{eqnarray*}
	This yields a bound depending only on $r, |a|, |b|, K_r, L_r$ as required.
\end{proof}

\subsection{Finding a good curve}
In the following statement we consider $n\in{\mathbb N}$ even, so that $q_{n} {\alpha} - p_n >0$.

\begin{prop} \label{prop smooth return domain}
	There is a sequence of increasing functions $\{G_r : {\mathbb R}_+^{2} \to {\mathbb R}_+\}_{r \geq 1}$ such that the following is true.

	Let $f$ be a pseudo-rotation in $\diff^\infty({\mathbb A}, \omega)$, let $F$ be a lift with $\rho(F) = \rho(f) = {\alpha} \in (0, 1) \setminus {\mathbb Q}$, and let $n \geq 2$ be an even integer.    Suppose $\gamma\subset{\mathbb A}$ is a smooth Brouwer curve for $f^{q_n}$ with a lift $\gamma^{\dagger}\subset\tilde{\mathbb A}$ such that the closed region $\Omega$ in $\tilde{\mathbb A}$ bounded by $(\gamma^\dagger, F^{q_n, -p_n}(\gamma^\dagger))$ admits an $(r, K_r)$-admissible coordinate
	\begin{eqnarray*}
				H:[0,1]^2\rightarrow\Omega
	\end{eqnarray*}
	for some $r \geq 1$.
	Suppose further that $\gamma_n\subset{\mathbb A}$ is an $a_{n+2}$-good smooth curve for $f^{q_{n+1}, -p_{n+1}}_H$,\footnote{ Recall that by Lemma \ref{prop. return map} we have $\rho(f^{q_{n+1}, -p_{n+1}}_H) = {\alpha}_{n+1}$ and $a_{n+2} = q({\alpha}_{n+1})$.} for which the closed region $\Omega_n\subset{\mathbb A}$ bounded by $(\gamma_n,f^{q_{n+1}, -p_{n+1}}_H(\gamma_n))$ admits an $(r, M_r)$-admissible coordinate.
	Then there exists a $q_{n+2}$-good smooth curve $\hat\gamma\subset{\mathbb A}$ for $f$, for which the closed region $\Omega_*\subset{\mathbb A}$ bounded by  $(f^{q_{n+1}}(\hat\gamma), \hat\gamma)$ admits a $(r, G_r( K_r,  M_r))$-admissible coordinate for $f^{q_{n+1}}$.
\end{prop}
\begin{proof}
	We set
	\begin{eqnarray*}
		S_0 = F^{q_n, -p_n},\qquad S = F^{q_{n+1}, -p_{n+1}}.
	\end{eqnarray*}
	As explained in Section \ref{Construction of a new pseudo-rotation}, we extend $H$ to a $C^\infty$-diffeomorphism of $\tilde {\mathbb A}$ with constant Jacobian by the formula
	\begin{eqnarray*}
	H T = S_0 H.
	\end{eqnarray*}
	By definition, we know that
	\begin{enumerate}
	\item $JH^{-1}S_0HJ = JTJ = T^{-1}$ on $\tilde {\mathbb A}$;
	\item $\tilde S := JH^{-1}S HJ$ commutes with $T$ on $\tilde {\mathbb A}$, and descends to $f^{q_{n+1}, -p_{n+1}}_H$. Moreover, we have $\rho(\tilde S) = \rho(f^{q_{n+1}, -p_{n+1}}_H) = {\alpha}_{n+1} \in (0, \frac{1}{2})$.
	\end{enumerate}

	Let $\tilde \gamma$ be an arbitrary lift of $\gamma_n$.
	As $ \gamma_n$ is a simple regular curve and is disjoint from $f^{q_{n+1}, -p_{n+1}}_H( \gamma_n)$, we know by item (1), (2) above that $\tilde \gamma$ is also a simple regular curve, and
	\begin{eqnarray} \label{eq ttilegammadisjointfromtildegamma}
	T^{-1}(\tilde \gamma) \cap \tilde \gamma  = \tilde S(\tilde \gamma) \cap \tilde \gamma = \emptyset.
	\end{eqnarray}
	We let $\gamma' = HJ(\tilde \gamma)$. Then $\gamma'$ is also a simple regular curve, and by \eqref{eq ttilegammadisjointfromtildegamma} and item (1), (2) above, we have
	\begin{eqnarray} \label{eq s0sgamma'disjoint}
	S_0( \gamma') \cap \gamma' = S(\gamma') \cap \gamma' = \emptyset.
	\end{eqnarray}
    Moreover, we can see that
    \begin{eqnarray} \label{eq orderofss0gamma'}
    S(\gamma') < \gamma' < S_0(\gamma')
    \end{eqnarray} by considering their boundary points on $B_0$ and $B_1$. We let $\hat \gamma = \pi(\gamma')$ (recall that $\pi$ is the canonical projection from $\tilde {\mathbb A}$ to ${\mathbb A}$).
	\begin{lemma}\label{lem simplecurve}
		The curve $\hat \gamma$ is a simple  regular curve connecting $B_0$ and $B_1$. Namely, it has no self-intersection.
	\end{lemma}
    \begin{proof}
    	Since $\hat \gamma = \pi(\gamma')$, and since $\gamma'$ is a simple regular curve in $\tilde {\mathbb A}$ connecting $B_0$ to $B_1$, it suffices to show that $\gamma'$ is disjoint from all its translates $T^k(\gamma')$ for $k\in{\mathbb Z}\backslash\{0\}$.   Since $\gamma'$ connects the two boundary components it is enough to show $\gamma'$ is disjoint from $T(\gamma')$.

    	By definition and by \eqref{relation pn qn}, we have
    	$T = S_0^{q_{n+1}}S^{-q_n}$. Then by \eqref{eq orderofss0gamma'}, we obtain that
    	$\gamma' < T(\gamma')$.
    	This completes the proof.
    \end{proof}
    By item (2) above, the region $\tilde \Omega_n$ bounded by $(\tilde \gamma, \tilde S(\tilde \gamma))$ is a lift of $\Omega_n$
    Moreover, the $(r, M_r)$-admissible coordinate for $\Omega_n$ lifts to a   $(r, M_r)$-admissible coordinate for $\tilde\Omega_n$.
We notice that the push forward $HJ(\tilde \Omega_n)$ is the region in $\tilde {\mathbb A}$ between $S(\gamma')$ and $\gamma'$.
Moreover, from the proof of Lemma \ref{lem simplecurve}, we see that
\begin{eqnarray*}
T^{-1}(\gamma') < S(\gamma') < \gamma'.
\end{eqnarray*}
	Thus the map $\pi$ induces a diffeomorphism from $HJ(\tilde \Omega_n)$ to the region in ${\mathbb A}$ between $f^{q_{n+1}}(\hat\gamma)$ and $\hat\gamma$,
	that is, to the region  $\Omega_*$.
	We conclude that the $(r, M_r)$-admissible coordinate for $\tilde{\Omega}_n$, after composing with $HJ$ and projecting, yield $(r, G_r( K_r, M_r))$-admissible coordinates for $\Omega_*$, for some functions $G_r$ as in the proposition.

	It remains to show that $\hat \gamma$ is a $q_{n+2}$-good curve. We divide the proof into two cases.
	\smallskip

	Case I: \
	Assume that there are integers $p$ and $0 < k < q_{n+2}$ such that
	$k{\alpha} + p < 0$ and
	\begin{eqnarray} \label{intersect 1}
	F^{k, p}( \gamma') \cap \gamma' \neq \emptyset.
	\end{eqnarray}
	We assume further that for any integers $p'$ and $0 < k' < q_{n+2}$ such that $k'{\alpha} + p' < 0$ and $F^{k', p'}( \gamma') \cap  \gamma' \neq \emptyset$, we have
	\begin{eqnarray*}
	k'{\alpha} + p' \leq k {\alpha} + p.
	\end{eqnarray*}

	$\bullet$
	First, we observe that either $q_{n+2} > k \geq q_{n+2} - q_n$ or
	\begin{eqnarray*}
	-\{q_n {\alpha}\} = p_n-q_n {\alpha}  < k {\alpha} + p < 0.
	\end{eqnarray*}
	Indeed, if $k < q_{n+2} - q_n$ and $p_n-q_n {\alpha}  >  k {\alpha} + p$, then we have $k + q_n < q_{n+2}$ and
	\begin{eqnarray*}
	k {\alpha} + p < (k+q_n){\alpha} + p-p_n < 0.
	\end{eqnarray*}
	In particular, the endpoints of $S_0F^{k, p}( \gamma')$ are on the left hand side of those of $\gamma'$.
	Then by \eqref{intersect 1} and $S_0F^{k, p}( \gamma') > F^{k,p}( \gamma')$, we have that
	\begin{eqnarray*}
	F^{k + q_n, -p_n + p}(\gamma' ) \cap  \gamma'  \neq \emptyset.
	\end{eqnarray*}
	This contradicts the choice of $k$.

	$\bullet$ If $q_{n+2} > k \geq q_{n+2} - q_n$, then by \eqref{eq s0sgamma'disjoint}, \eqref{intersect 1} and \eqref{q iteration}, we know that $k > q_{n+1}$. We notice that
	\begin{eqnarray*}
	k {\alpha} + p < (k-q_{n+1}){\alpha} + p + p_{n+1} < 0.
	\end{eqnarray*}
		In particular, the endpoints of $S^{-1}F^{k, p}( \gamma')$ are on the left hand side of those of $\gamma'$.
	Then by \eqref{intersect 1} and $S^{-1} F^{k, p}( \gamma') > F^{k,p}( \gamma')$, we have that
	\begin{eqnarray*}
	F^{k-q_{n+1}, p + p_{n+1}}( \gamma') \cap  \gamma' \neq \emptyset.
	\end{eqnarray*}
	This contradicts the choice of $k$.

	$\bullet$ 	If $-\{q_n {\alpha}\}   < k {\alpha} + p < 0$, then we have
	\begin{eqnarray*}
	k \in \{q_{n+1}, \cdots, a_{n+2}q_{n+1}\}.
	\end{eqnarray*}
	If $k = i q_{n+1}$ with $i > 1$, then we have
	\begin{eqnarray*}
	-1 < i (q_{n+1} {\alpha} - p_{n+1}) < (i-1)(q_{n+1} {\alpha} - p_{n+1}) = (k-q_{n+1}) {\alpha} -(i-1)p_{n+1} <  0.
	\end{eqnarray*}
	Then we must have $p = -ip_{n+1}$, and the endpoints of $S^{-1}F^{k, p}( \gamma')$ are on the left hand side of those of $\gamma'$.
	Then by \eqref{intersect 1} and $S^{-1} F^{k, p}( \gamma') > F^{k,p}( \gamma')$, we have that
	\begin{eqnarray*}
	F^{k-q_{n+1}, -(i-1)p_{n+1}}( \gamma') \cap  \gamma' \neq \emptyset.
	\end{eqnarray*}
	If $k = q_{n+1}$, we would have
	\begin{eqnarray*}
	F^{q_{n+1}}(\gamma') \cap ( \gamma' + {\mathbb Z}) = \emptyset.
	\end{eqnarray*}
	Both cases contradict the choice of $k$.\\

		Case II: \
	Assume that there are integers $p$ and $0 < k < q_{n+2}$ such that $k {\alpha} + p > 0$ and
	\begin{eqnarray} \label{intersect 2}
	F^{k, p}( \gamma') \cap \gamma' \neq \emptyset.
	\end{eqnarray}
	We  assume further that for any integers $p'$ and $0 < k' < q_{n+2}$ such that
	$k'{\alpha} + p' > 0$ and $F^{k', p'}( \gamma') \cap  \gamma' \neq \emptyset$, we have
	\begin{eqnarray*}
	k'{\alpha} + p' \geq k {\alpha} + p.
	\end{eqnarray*}

		$\bullet$
	First, we observe that either $q_{n} \geq k > 0$ or
	\begin{eqnarray*}
	\{q_n {\alpha}\} = q_n {\alpha} - p_n  > k {\alpha} + p > 0.
	\end{eqnarray*}
	Indeed, if $k >   q_n$ and $ q_n {\alpha} - p_n  <  k {\alpha} + p$, then we have $q_{n+2} > k - q_n  > 0$ and
	\begin{eqnarray*}
	k {\alpha} + p  >  (k-q_n){\alpha} + p + p_n >  0.
	\end{eqnarray*}
	In particular, the endpoints of $S_0^{-1}F^{k, p}( \gamma')$ are on the right hand side of those of $\gamma'$.
	Then by \eqref{intersect 2} and $F^{k - q_n, p_n + p}( \gamma' ) = S_0^{-1}F^{k, p}( \gamma')< F^{k,p}( \gamma')$, we have that
	\begin{eqnarray*}
	F^{k - q_n, p_n + p}( \gamma' ) \cap  \gamma'  \neq \emptyset.
	\end{eqnarray*}
	This contradicts the choice of $k$.

		$\bullet$ If $q_{n} \geq k  > 0$, then by \eqref{eq s0sgamma'disjoint} and \eqref{intersect 2}, we have $q_n > k$ and consequently $k + q_{n+1} < q_{n+2}$.
		Moreover we notice that
	\begin{eqnarray*}
	k {\alpha} + p  >  (k+q_{n+1}){\alpha} + p - p_{n+1}  >  0.
	\end{eqnarray*}
		In particular, the endpoints of $S F^{k, p}( \gamma')$ are on the right	 hand side of those of $\gamma'$.
	By \eqref{intersect 2} and $S F^{k, p}( \gamma') < F^{k,p}( \gamma')$, we have  that
	\begin{eqnarray*}
	F^{k+q_n, p - p_n}( \gamma') \cap \gamma' \neq \emptyset.
	\end{eqnarray*}
	This contradicts the choice of $k$.

		$\bullet$ 	If $\{q_n {\alpha}\}   >  k {\alpha} + p >  0$ (and $0 < k < q_{n+2}$), then it is straightforward to verify that
	\begin{eqnarray*}
	(k,p) \in \{(q_{n} + iq_{n+1}, p_n + i p_{n+1} ) \mid 0 <  i < a_{n+2}\}.
	\end{eqnarray*}
	By the hypothesis that $\gamma_n$ is a $a_{n+2}$-good curve,
	we have
	\begin{eqnarray*}
	T^{-1}\tilde S^i(\tilde \gamma) \cap \tilde \gamma = \emptyset \quad  \forall 1 \leq i < a_{n+2}.
	\end{eqnarray*}
	By $T^{-1} = JH^{-1} S_0 HJ$, $\tilde S = JH^{-1}S HJ$ and $\gamma' = HJ(\tilde \gamma)$, we obtain
	\begin{eqnarray*}
	F^{q_n + i q_{n+1}, -p_n - ip_{n+1}}(\gamma') \cap \gamma' = 	S_0S^{i}(\gamma') \cap \gamma' = \emptyset
	\end{eqnarray*}
	for all integer $0 < i < a_{n+2}$. This again gives a contradiction.

	In summary, we see that for all integer $0 < i < q_{n+2}$,
	\begin{eqnarray*}
		F^{i}(\gamma') \cap (\gamma' + {\mathbb Z}) = \emptyset.
	\end{eqnarray*}
	 Hence $\hat \gamma$ is $q_{n+2}$-good.   This completes the proof of Proposition \ref{prop smooth return domain}.
\end{proof}

We can now prove the following:

\begin{cor}\label{cor smooth return domain}
	For each integer  $r \geq 1$,
	there exist increasing functions $P_r:  {\mathbb R}^2_{+} \to {\mathbb R}_+$ and  $ W_r : {\mathbb N} \times  {\mathbb R}_+ \to {\mathbb R}_+$  such that  the following is true.
	Suppose that there is an odd integer $n\geq 3$ such that
	\begin{eqnarray} \label{eq qnqn+1qn+2}  \\
q_{n} > P_r(K_{r+2}, q_{n-1}), 	\quad q_{n+1} > P_r(K_{r+2}, q_{n}) \quad \mbox{and} \quad q_{n+2} > P_{r}(K_{r+2}, q_{n+1}) \nonumber
	\end{eqnarray}
	where $\{ q_{k} \}_{k \geq 0}$ is the sequence of denominators associated to some ${\alpha}\in(0,1)\backslash{\mathbb Q}$.
	Then for any $\{K_k\}_{k \geq 1}$-smooth pseudo-rotation $f\in \diff^\infty({\mathbb A}, \omega)$ with $\rho(f) = {\alpha}$ has a $q_{n+1}$-good smooth curve
	$\gamma$ such that the closed region in ${\mathbb A}$ bounded by $(  f^{q_n}(\gamma), \gamma)$ admits a
	$(r,  W_r( q_{n+1},   K_{r+2}))$-admissible coordinate for $f^{-q_n}$.
\end{cor}

\begin{comment}
\green{
Note: If one substitutes $n=0$ then the statement is almost the same as Corollary \ref{cor main2}, except that the estimates here require an extra derivative - $K_{r+2}$ appears instead of $K_{r+1}$.
Indeed, let's compare the assumptions when $n=0$ with those of Corollary \ref{cor main2}: When $n=0$ the inequalities \eqref{eq qnqn+1qn+2} are equivalent to
\begin{eqnarray*}
		{\alpha}^{-1}>P_r(K_{r+2},1)\quad \mbox{and} \quad {\alpha}^{-1}          \mathcal G({\alpha})^{-1} > P_{r}(K_{r+2}, {\alpha}^{-1})
\end{eqnarray*}
for a slightly modified increasing function $P_r$, which is equivalent to
\begin{eqnarray}\label{E: a aGa}
		{\alpha}<C_r(K_{r+2}^{-1},1)\quad \mbox{and} \quad {\alpha}          \mathcal G({\alpha}) < C_{r}(K_{r+2}^{-1}, {\alpha})
\end{eqnarray}
for some increasing function $C_r$.   We can compare this to the assumption ${\alpha}\in\mathcal L_r(K_{r+1})$ in Corollary \ref{cor main2}, namely:
\begin{eqnarray*}
		A_0({\alpha}          \mathcal G({\alpha}), K^{q({\alpha})}_{r+1})  <  K^{-q({\alpha})}_{r+1} C'_r({\alpha}^{-1},   K_{r+1})^{-1}.
\end{eqnarray*}
Since ${\alpha}\mapsto K^{-q({\alpha})}$, $A_0$ and $C'_r$ are all increasing, this condition can be rewritten in the form ${\alpha}          \mathcal G({\alpha}) < C'''_{r}(K_{r+2}^{-1}, {\alpha})$
for some increasing function $C'''_r$, which is therefore comparable to the second inequality in \eqref{E: a aGa}.   Now let's compare the conclusions:
When $n=0$ in Corollary \ref{cor smooth return domain} the conclusion should refer to the region $(\gamma, f^{q_n}(\gamma))$ (since $n=0$ is even).
That is, there exists a $q_1=q({\alpha})$-good Brouwer curve $\gamma$ such that the region $(\gamma, f^{q_0}(\gamma))=(\gamma, f(\gamma))$ admits an
$(r,  W_r( q({\alpha}),   K_{r+2}))$-admissible coordinate.   Since $q({\alpha})=\lfloor1/{\alpha}\rfloor$ this is equivalent to $(r,  W'_r({\alpha}^{-1},   K_{r+2}))$-admissible coordinates,
for some increasing function $W'_r$.   If we replaced $K_{r+2}$ here by $K_{r+1}$ this would be comparable to the conclusion in Corollary \ref{cor main2}.

\end{comment}

\begin{proof}
	Fix $r \geq 1$.  Let $\{L_k: {\mathbb N} \times {\mathbb R}_+ \to {\mathbb R}_+ \}_{k \geq 1}$ be a sequence of  increasing functions independent of $f$ such that for each integer $m \geq 0$, $f^m$ is $\{ L_k(m, K_k) \}_{k \geq 1}$-smooth.
    Let us abbreviate
	\begin{eqnarray*}
	L'_k := L_k(q_{n-1},  K_k) \qquad \forall k \geq 1.
	\end{eqnarray*}
	For each $n \geq 2$ we have $\rho(f^{q_{n-1}}) = \beta_{n-1} \in (0,1/2)$.
	If \eqref{eq qnqn+1qn+2} holds and $P_r$ is
	chosen appropriately, then
	\begin{eqnarray}\label{condition_beta}
						\beta_{n-1}\in\mathcal L(L'_1).
	\end{eqnarray}
	This allows us to apply Lemma \ref{lem brouwer curve compactness} to $f^{q_{n-1}}$.
	By Lemma \ref{lem brouwer curve compactness}, there is a smooth Brouwer curve $\gamma$ for $f^{q_{n-1}}$ for which the closed region
	$\mathcal R'_{n-1}\subset{\mathbb A}$ bounded by $(\gamma, f^{q_{n-1}}(\gamma) )$ admits a $(r+1, Y_{r+1})$-admissible coordinate $H' : [0,1]^2 \to \mathcal R'_{n-1}$ 	where
	\begin{eqnarray} \label{def yr}
	Y_{r+1}  :=   C'_{r+1}( \beta_{n-1}^{-1},   L'_{r + 2}).
	\end{eqnarray}
 We lift $H'$ to an admissible coordinate $H: [0,1]^2 \to \mathcal R_{n-1}$ where $\mathcal R_{n-1}$ is a lift of $\mathcal R'_{n-1}$ in $\tilde{\mathbb A}$.  More precisely,
 $\mathcal R_{n-1}$ is a connected component of $\pi^{-1}(\mathcal R'_{n-1})$.
As in Section \ref{Construction of a new pseudo-rotation}, we extend $H$ to a diffeomorphism of $\tilde {\mathbb A}$ satisfying
\begin{eqnarray*}
						HT=F^{q_{n-1},-p_{n-1}}H
\end{eqnarray*}
and set $f_n := f^{q_{n}, -p_{n}}_{H}:{\mathbb A}\rightarrow{\mathbb A}$.   That is, $f_n$ is the projection of
$F^{q_{n}, -p_{n}}_{H}:=(HJ)^{-1}F^{q_{n}, -p_{n}}(HJ):\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}$.
By Lemma \ref{prop. return map}, $f_n$ is a pseudo-rotation in $\diff^\infty({\mathbb A}, \omega)$ with $\rho(f_n) = \beta_n/\beta_{n-1}={\alpha}_{n}$ such that
\begin{eqnarray} \label{eq crnormoffn_0}
\norm{f_n}_{\diff^{r+1}} \leq  E_{r+1}(q_{n}, p_n,  K_{r+1},  Y_{r+1}).
\end{eqnarray}
By \eqref{def yr}, and using that
\begin{eqnarray*}
K_{r+1} \leq K_{r+2}, \quad
\max( p_n, \beta_{n-1}^{-1}) \leq 2q_n
\end{eqnarray*}
we can rewrite \eqref{eq crnormoffn_0} as
\begin{eqnarray} \label{eq crnormoffn}
\norm{f_n}_{\diff^{r+1}}  < V_{r+1}(q_{n}, K_{r+2})
\end{eqnarray}
where $ V_{r+1} : {\mathbb N} \times {\mathbb R}_+ \to (1, \infty)$ is an increasing function independent of $f$.

Now we choose $P_r: {\mathbb R}^2_+ \to {\mathbb R}_+$ to be any increasing function that is sufficiently large that the following conditions are fulfilled:
	  \begin{eqnarray}
	  P_r(D,y) &>& 2y\rho_*(V_{r+1}(y, D))^{-1}  \label{eq p property 0}\\
	  \frac{2y}{P_r(D,y)} &<& C''\left( \frac{1}{2y},  V_{r+1}(y, D)^{-1} \right),\label{eq p property 1}
	 \end{eqnarray}
	for all $D,y>0$, where $\rho_*$ is the decreasing function introduced
	in \eqref{def a}.   We moreover choose $P_r$ sufficiently large that whenever $x>P_r(D,y)$ there holds
	 \begin{eqnarray}
	 \label{eq p property 2}
	 \ A_0\left(\frac{2y}{x},  V_{r+1}(y, D)^{2y} \right) &<&  V_{r+1}(y, D)^{-2y} C'_r(2y,  V_{r+1}(y, D))^{-1}
	 \end{eqnarray}
	 for all $D,y>0$.   This we can arrange because $\lim_{t\rightarrow 0}A_0(t,\cdot)=0$.

 Assume now that \eqref{eq qnqn+1qn+2} holds for a fixed odd integer $n\geq3$.
 By \eqref{eq p property 0} and then the second inequality in \eqref{eq qnqn+1qn+2}, we have
 \begin{eqnarray*}
 2 q_n &<& \rho_*(V_{r+1}(q_n,K_{r+2}))P_{r}(K_{r+2},q_n) < \rho_*(V_{r+1}(q_{n}, K_{r+2}))q_{n+1}
 \end{eqnarray*}
and therefore by \eqref{eq anupperlower}
 \begin{eqnarray} \label{eq ansmall}
 {\alpha}_n < \frac{2 q_n}{q_{n+1}}  &<& \rho_*(V_{r+1}(q_{n}, K_{r+2})).
 \end{eqnarray}
Also, by \eqref{eq anupperlower}, the third inequality in \eqref{eq qnqn+1qn+2}, and \eqref{eq p property 1}, we obtain
\begin{eqnarray}
          \mathcal G({\alpha}_n) = {\alpha}_{n+1} &<& \frac{2q_{n+1}}{q_{n+2}} < \frac{2q_{n+1}}{P_r(K_{r+2},q_{n+1})} \\
				&<& C''\left(\frac{1}{2q_{n+1}},  V_{r+1}(q_{n+1}, K_{r+2})^{-1}\right)  \nonumber \\
&<&  C''({\alpha}_n,V_{r+1}(q_{n}, K_{r+2})^{-1}) \label{eq gansmall}
\end{eqnarray}
where the last inequality uses the monotonicity of $C''$.   We claim that
\begin{eqnarray} \label{eq rhofninlr}
\rho(f_n) = {\alpha}_{n} \in \mathcal L_r(V_{r+1}(q_{n}, K_{r+2}))
\end{eqnarray}
where $\mathcal L_r$ is as in Definition \ref{def Lr}.   Equivalently, we show that
\begin{eqnarray}\label{eq rhofninlr 1}
	\\
	A_0\left({\alpha}_n           \mathcal G({\alpha}_n), V_{r+1}(q_{n}, K_{r+2})^{q({\alpha}_n)}\right) < V_{r+1}(q_{n}, K_{r+2})^{-q({\alpha}_n)} C'_r\Big({\alpha}_n^{-1},  V_{r+1}(q_n, K_{r+2})\Big)^{-1}.\nonumber
\end{eqnarray}
Using \eqref{eq anupperlower} we have ${\alpha}_n           \mathcal G({\alpha}_n)={\alpha}_n{\alpha}_{n+1}< 4q_{n}/q_{n+2}\leq 2q_{n+1}/q_{n+2}$ and $q({\alpha}_n)\leq 2q_{n+1}$
and ${\alpha}_n^{-1}<2q_{n+1}/q_n\leq 2q_{n+1}$.   Therefore by the monotonicity of $A_0$,  and by \eqref{eq p property 2} together with the third inequality in \eqref{eq qnqn+1qn+2}, we obtain
\begin{eqnarray*}
A_0\left({\alpha}_n           \mathcal G({\alpha}_n), V_{r+1}(q_{n}, K_{r+2})^{q({\alpha}_n)}\right) &<& A_0\left(\frac{2q_{n+1}}{q_{n+2}},  V_{r+1}(q_{n+1}, K_{r+2})^{2q_{n+1}}\right) \\
&<&  V_{r+1}(q_{n+1}, K_{r+2})^{-2q_{n+1}} C'_r( 2q_{n+1},  V_{r+1}(q_{n+1},K_{r+2}) )^{-1}.
\end{eqnarray*}
Now inequality \eqref{eq rhofninlr 1} follows from the monotonicity of $V_{r+1}$ and $C'_r$, and because ${\alpha}_n^{-1}<2q_{n+1}/q_n\leq 2q_{n+1}$ from
\eqref{eq anupperlower}.
This proves \eqref{eq rhofninlr}.

Combining \eqref{eq rhofninlr} with \eqref{eq crnormoffn} we see that $f_n$ satisfies the hypotheses of Proposition \ref{prop. main}, and
therefore $f_n$ has a $q(\rho(f_n))$-good Brouwer curve $\gamma_n$ say.   By \eqref{eq a n} $q(\rho(f_n))=q({\alpha}_n)=a_{n+1}$.   Thus
$\gamma_n$ is an $a_{n+1}$-good Brouwer curve for $f_n$.   Moreover, by Lemma \ref{lem brouwer curve compactness}, the region in ${\mathbb A}$ bounded by $(\gamma_n, f_n(\gamma_n))$  admits an $(r,C'_r({\alpha}_n^{-1},\hat{K}_{r+1}))$-admissible coordinate, provided $\hat{K}_{r+1}\geq\|f_n\|_{\diff^{r+1}}$
and $\rho(f_n)={\alpha}_n\in\mathcal L(\hat{K})$ where $\hat{K}\geq\|f_n\|_{\diff^{1}}$.    By \eqref{eq rhofninlr} and \eqref{eq crnormoffn} we may
take $\hat{K}_{r+1}=\hat{K}=V_{r+1}(q_n, K_{r+2})$ and conclude that
the region in ${\mathbb A}$ bounded by $(\gamma_n, f_n(\gamma_n))$ admits a $(r, U_r)$-admissible coordinate where
	\begin{eqnarray*}
	U_r := C'_r({\alpha}_{n}^{-1}, V_{r+1}(q_n, K_{r+2})).
	\end{eqnarray*}
	We define $W_r$ by
	\begin{eqnarray*}
	W_r(q, K)
	:= G_r(C'_{r}(2q, L_{r+1}(q, K)), C'_r(2q, V_{r+1}(q, K))).
	\end{eqnarray*}
	By \eqref{eq continuefraction} and \eqref{eq anupperlower}, we have $          \beta_{n-1}^{-1}, {\alpha}_n^{-1} < 2q_{n+1}$. Then by $K_{r+2} \geq K_{r+1}$, we have
	\begin{eqnarray*}
&&	C'_{r}(2q_{n+1}, L_{r+1}(q_{n+1}, K_{r+2})) > Y_{r} \\
&\mbox{and }&	C'_r(2q_{n+1}, V_{r+1}(q_{n+1}, K_{r+2})) > U_r.
	\end{eqnarray*}
    Thus we have
	\begin{eqnarray*}
	W_r(q_{n+1}, K_{r+2}) \geq G_r(Y_{r}, U_r).
	\end{eqnarray*}
	Then by Proposition \ref{prop smooth return domain}, there is a $q_{n+1}$-good smooth curve of $f$, denoted by $\hat\gamma$, such that  the region in ${\mathbb A}$ bounded by $(f^{q_{n}}(\hat\gamma), \hat\gamma)$ admits a $(r, W_r(q_{n+1}, K_{r+2}))$-admissible coordinate for $f^{-q_n}$.
\end{proof}

\section{Construction of approximants}

This section is mostly occupied by the proof of the following theorem, from which
the main result of this paper, Theorem \ref{main thm F = closure O}, will then follow easily.

\begin{thm}\label{thm. approximation}
	 For each $(r_0,M,\epsilon) \in {\mathbb Z}_{\geq 2} \times {\mathbb N}\times(0,1]$, there is an increasing function
	 \begin{eqnarray*}
	 			P=P_{r_0,\epsilon,M}:{\mathbb N}\rightarrow {\mathbb R}_+
	\end{eqnarray*}
	 so that for any pseudo-rotation $f\in \diff^\infty({\mathbb A}, \omega)$ with
	\begin{eqnarray}\label{Cr bounds on f}
	 		 \norm{f}_{\diff^{r_0+2}({\mathbb A})} < M,
	\end{eqnarray}
	whose rotation number $\rho(f) = {\alpha} \in (0,1)\setminus {\mathbb Q}$ satisfies the property that there exists an odd integer  $n\geq 3$ for which
	\begin{eqnarray} \label{eq qn+2qn+1qnP}
	q_{n} > P(q_{n-1}),  \quad q_{n+1} > P(q_{n}) \quad \mbox{and} \quad		q_{n+2} > P(q_{n+1}),
	\end{eqnarray}
	then there exists $h_0 \in \diff^\infty({\mathbb A}, \omega)$ with
	\begin{eqnarray}\label{epsilon close}
	d_{\diff^{r_0 - 1}({\mathbb A})}(h_0 R_{{\alpha}}  h_0^{-1},f)< \epsilon.
	\end{eqnarray}
\end{thm}
\begin{proof}

	Fix some $(r_0,M, \epsilon)$ in ${\mathbb Z}_{\geq 2}\times {\mathbb N} \times (0,1]$.    Let $f\in\diff^\infty({\mathbb A},\omega)$ denote a pseudo-rotation and set ${\alpha} = \rho(f)$.

		It will be convenient to use the following notation: for an increasing function
		$S:{\mathbb N}\to{\mathbb N}$ we define
		\begin{align}
				          \mathcal C(S) &:= \Big\{ \theta \in (0,1)\backslash{\mathbb Q}\ \mid \ \exists n\in{\mathbb N}\mbox{ odd, so that }\\
						&\qquad q_{n}(\theta)>S(q_{n-1}(\theta)), \ q_{n+1}(\theta)>S(q_{n}(\theta)),		\			 q_{n+2}(\theta)>S(q_{n+1}(\theta)) \Big\}   \nonumber
		\end{align}
		where $\{ p_n(\theta)/q_n(\theta) \}_{n \geq 0}$ is the sequence of continued fractions of $\theta$ introduced in  Section \ref{sec prelim}.
		Our successive restrictions on the rotation number ${\alpha}$ will take the form:
		\begin{eqnarray}\label{conditions}
											{\alpha}\in           \mathcal C(S_i)
		\end{eqnarray}
		for a finite collection of functions $S_1, S_2,\ldots$ to be determined.
		We will set
		\begin{eqnarray*}
						 P:=\max_{i}S_i
		\end{eqnarray*}
		and then when ${\alpha}\in          \mathcal C(P)$, all conditions in (\ref{conditions}) will be met.

	    For our first condition on the rotation number, set
		\begin{eqnarray} \label{eq qn+2pmqn+1}
		S_1:=P_{r_0}(M,\cdot)
		\end{eqnarray}
		where $P_{r_0}:{\mathbb R}^2_{+} \to {\mathbb R}_+ $ is defined in Corollary \ref{cor smooth return domain}.
		From now on we assume
		\begin{eqnarray*}
				{\alpha}\in           \mathcal C(S_1).
		\end{eqnarray*}

		Fix any odd $n\geq 3$ for which
		\begin{eqnarray}\label{condition on n}
				q_{n}>S_1(q_{n-1}),  \quad	q_{n+1}>S_1(q_{n}) \quad \mbox{and} \quad	q_{n+2}>S_1(q_{n+1}).
		\end{eqnarray}
		Then by Corollary \ref{cor smooth return domain}, $f$ has a $q_{n+1}$-good Brouwer curve $\gamma\subset{\mathbb A}$ for which
		the closed region
		\begin{eqnarray*}
					          \mathcal D_0\subset{\mathbb A}
		\end{eqnarray*}
		bounded by $(f^{q_n}(\gamma),  \gamma)$
		admits a $(r_0,  \hat W_{r_0}    \color{black})$-admissible coordinate for $f^{-q_n}$, where
	\begin{eqnarray*}
	  \hat W_{r_0}    \color{black}  :=   W_{r_0}(  q_{n+1}, K_{r_0+2}).
	\end{eqnarray*}
	Recall that $\gamma$ is $q_{n+1}$-good for $f$ means that the curves  $f^i(\gamma)$, $0 \leq i \leq q_{n+1}-1$ are mutually disjoint.
    Then it is clear that the curves
	\begin{eqnarray} \label{boundary curves}
	f^{q_n}(\gamma), \cdots, f^{q_n + q_{n+1} - 1}(\gamma)
	\end{eqnarray}
	are mutually disjoint as well.   In particular $f^{q_{n+1}}(\gamma)$ and $f^{q_n}(\gamma)$ are disjoint and so we can consider
	the closed region
	\begin{eqnarray*}
			          \mathcal D\subset {\mathbb A}
	\end{eqnarray*}
	bounded by $(f^{q_{n}}(\gamma),f^{q_{n+1}}(\gamma))$.

	\begin{claim} \label{claim covering}
	The annulus ${\mathbb A}$ is covered by the regions
	\begin{eqnarray*}
	          \mathcal D, f(          \mathcal D), \cdots, f^{q_{n}-1}(          \mathcal D), f^{q_n}(          \mathcal D_0), \cdots, f^{q_{n+1}-1}(          \mathcal D_0)
	\end{eqnarray*}
	with mutually disjoint interiors. Moreover, the intersection of any two such neighboring regions equals one of the curves in
	\eqref{boundary curves}.
	\end{claim}
\begin{proof}
	The restriction of $f$ to the boundary circle $B_0$ is an orientation preserving homeomorphism with rotation number ${\alpha}$.
	It is then a well-known fact that for any fixed $x_0\in B_0$ the intervals
	\begin{eqnarray*}
	& [f^{q_{n}  + i}(x_0), f^{q_{n+1} + i}(x_0)], \quad 0 \leq i \leq {q_n-1}, \\
	\mbox{ and } &  [f^{q_n + j}(x_0), f^{j}(x_0)], \quad q_n \leq j \leq q_{n+1}-1
	\end{eqnarray*}
	together form a covering for $B_0$ with mutually disjoint interiors.   In particular we may take
	$x_0 := \gamma \cap B_0$.   An analogous statement holds for the other boundary component $B_1$.
	The claim then follows from the disjointness of the curves in \eqref{boundary curves} and our definition of simple regular curve.
\end{proof}

		Fix a lift $\tilde \gamma$ of $\gamma$ to $\tilde {\mathbb A}$.  Let $F:\tilde{\mathbb A}\to\tilde{\mathbb A}$ be the unique lift of $f$ satisfying
		$\rho(F) = \rho(f) \in(0, 1)$.    Recall the notation from section \ref{S:gluing} that $F^{a,b}=T^bF^a$ for all $a,b\in{\mathbb Z}$.
	We define $\tilde           \mathcal D_0\subset\tilde{\mathbb A}$ to be the region bounded by
		\begin{eqnarray*}\label{lift D_0}
						\big( F^{q_n, -p_n}(\tilde \gamma), \tilde \gamma \big)
		\end{eqnarray*}
		and $\tilde           \mathcal D\subset\tilde{\mathbb A}$ to be the region bounded by
		\begin{eqnarray*}\label{lift D}
						\big( F^{q_n, -p_n}(\tilde \gamma), F^{q_{n+1}, -p_{n+1}}(\tilde \gamma) \big).
		\end{eqnarray*}

		We claim that
		$T^{-p_n}F^{q_n}(\tilde\gamma)$ is contained in the fundamental domain of $\pi$ bounded by $T^{-1}\tilde\gamma$ and $\tilde\gamma$. Indeed, this follows from comparing the order of their endpoints, and by the disjointness of their projections $f^{q_n}(\gamma)$ and $\gamma$.
		Thus $\pi$ restricts to a diffeomorphism from  $\tilde          \mathcal D_0$ to $          \mathcal D_0$.
		Similarly, $\pi$ restricts to a diffeomorphism from  $\tilde          \mathcal D$ to $          \mathcal D$ as well.

		The following $q_{n+1}$ regions have disjoint interiors:
\begin{eqnarray} \label{eq defoftildeb}
\tilde           \mathcal B_i := \begin{cases}
	F^{i}(\tilde           \mathcal D) & 0 \leq i \leq q_n-1, \\
	F^{i}(\tilde           \mathcal D_0) & q_n \leq i \leq q_{n+1}-1.
\end{cases}
\end{eqnarray}
By Claim \ref{claim covering}, $\tilde {\mathbb A}$ is covered by the union of the following regions with mutually disjoint interiors:
\begin{eqnarray} \label{def tcb j}
T^{-k} \tilde           \mathcal B_{j}, \quad k \in {\mathbb Z}, \ 0 \leq j \leq q_{n+1}-1.
\end{eqnarray}
    		The intersection of any two neighboring regions in \eqref{def tcb j} therefore equals to one of the following curves
\begin{eqnarray} \label{def tgamma}
T^{-k}F^{j}(\tilde \gamma), \quad k \in {\mathbb Z},\ q_n \leq j \leq q_n + q_{n+1}-1.
\end{eqnarray}
We denote by $\Gamma$ the union of the curves in \eqref{def tgamma}. The region
\begin{eqnarray} \label{def u0}
U_0 := \bigcup_{i=0}^{q_{n+1}-1}{\rm int}(\tilde           \mathcal B_i)
\end{eqnarray}
satisfies that:
\begin{enumerate}
\item $T^j(U_0) \cap T^k(U_0) = \emptyset$ for any $j \neq k \in {\mathbb Z}$;
\item $\cup_{k \in {\mathbb Z}} T^k(U_0) = \tilde {\mathbb A} \setminus \Gamma$.
\end{enumerate}

By Corollary \ref{cor smooth return domain}, there is an admissible coordinate
$\phi$ (see Definition \ref{def ad coord}) from a neighborhood of $ [0,1]^2$ in $\tilde {\mathbb A}$ to a neighborhood of $          \mathcal D_0$, satisfying \eqref{item 1}-\eqref{item 4} for $(f^{-q_n}, f^{q_n}(\gamma))$ in place of $(f, \gamma)$, and
	\arys
	\norm{\phi}_{\diff^{r_0}} &<&  \hat W_{r_0}, \\

	\phi\circ T (x) &=&   f^{-q_n} \circ\phi(x) \quad \forall x \in V
	\end{eqnarray*}
where $V$ is some neighborhood of $\{0\} \times [0,1]$ in $\tilde {\mathbb A}$.
As we have seen in Remark \ref{rema unique continuation}, there is no loss of information by regarding $\phi$ as a mapping from $[0,1]^2$ to $          \mathcal D_0$.
Since the restriction $\pi:\tilde          \mathcal D_0\rightarrow          \mathcal D_0$ is a diffeomorphism we may set
	\begin{eqnarray*}
				\psi := \pi^{-1}\circ\phi
	\end{eqnarray*}
which therefore gives us a $C^\infty$-smooth diffeomorphism that
extends to a  $C^\infty$-smooth diffeomorphism with constant Jacobian from a neighborhood of $ [0,1]^2$ in $\tilde {\mathbb A}$ to a neighborhood of $\tilde           \mathcal D_0$ satisfying
	\begin{eqnarray} \label{eq normofpsiyr0}
	\norm{\psi}_{\diff^{r_0}} &<&  \hat W_{r_0},    \color{black} \\
	\label{eq t psi}
	\psi\circ T (x) &=&   F^{-q_n, p_n}\circ \psi(x) \quad \forall x \in V.
	\end{eqnarray}
There is a unique extension to a $C^\infty$-diffeomorphism
    with constant Jacobian to the whole strip $\tilde {\mathbb A}$, which we still denote by
    \begin{eqnarray*}
    				\psi:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}
    \end{eqnarray*}
    and which satisfies
    \begin{eqnarray}\label{commuting condition}
    				\psi\circ T = F^{-q_n, p_n}\circ\psi.
    \end{eqnarray}

    To summarise thus far, we have proven the following:
    \begin{lemma}\label{S1}
	 For each $(r_0,M) \in {\mathbb Z}_{\geq 2} \times {\mathbb N}$ there is an increasing function
	 \begin{eqnarray*}
	 			S_1=S_1(r_0,M):{\mathbb N}\rightarrow {\mathbb R}_+
	\end{eqnarray*}
	 so that for any pseudo-rotation $f\in \diff^\infty({\mathbb A}, \omega)$ satisfying \eqref{Cr bounds on f}
	whose rotation number $\rho(f) = {\alpha}$ satisfies the property that there exists an odd integer $n \geq 3$ for which
	\begin{eqnarray*}
		q_{n} > S_1(q_{n-1}),  \quad
		q_{n+1} > S_1(q_{n}) \quad \mbox{and} \quad	q_{n+2} > S_1(q_{n+1}),
	\end{eqnarray*}
	there exists a $q_{n+1}$-good Brouwer curve $\gamma\subset{\mathbb A}$, so that for any lift $\tilde\gamma\in\tilde {\mathbb A}$,
	there exists a $C^\infty$-diffeomorphism,
    \begin{eqnarray*}
    				\psi:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A}
    \end{eqnarray*}
    	  with constant Jacobian which satisfies
	\begin{align}
	\qquad\qquad \qquad\qquad\norm{\psi}_{\diff^{r_0}([0,1]^2)} & <  \hat W_{r_0}     \color{black} :=   W_{r_0}(  q_{n+1}, M)\label{eq normofpsiyr2}
	\intertext{where $W_{r_0}$ is the function produced by Corollary \ref{cor smooth return domain}, and  }
	\psi\circ T & =  F^{-q_n, p_n}\circ \psi  \label{eq t psi 2}
	\end{align}
	on the whole of $\tilde {\mathbb A}$, where $F$ is the unique lift of $f$ with
	rotation number in $(0,1)$.   Moreover $\psi$
	maps $[0,1]^2$ onto the region $\tilde          \mathcal D_0$ bounded by $\big(\tilde F^{q_{n},-p_{n}}(\tilde\gamma), \tilde\gamma \big)$.
\end{lemma}
	To continue our proof of Theorem \ref{thm. approximation} we require the following:

    \begin{lemma}\label{S2}
     For each $(r_0,M) \in {\mathbb Z}_{\geq 2} \times {\mathbb N}$,
	 there is an increasing function
	 \begin{eqnarray*}
	 		Q_{r_0, M}: {\mathbb R}_+^{2} \to {\mathbb R}_+\quad\mbox{ with }\quad \lim_{t \to 0}Q_{r_0, M}(t,\cdot) \equiv 0,
	\end{eqnarray*}
	 and   there is an increasing function
	 \begin{eqnarray*}
	 			S_2=S_2(r_0, M):{\mathbb N}\rightarrow {\mathbb R}_+
	\end{eqnarray*}
	  with $S_2\geq S_1$,  so that for any pseudo-rotation $f\in \diff^\infty({\mathbb A}, \omega)$ satisfying \eqref{Cr bounds on f}
	whose rotation number $\rho(f) = {\alpha}$ satisfies the property that there exists an odd integer  $n\geq 3$ for which
	\begin{eqnarray*}
	q_{n} > S_1(q_{n-1}), \quad q_{n+1} > S_1(q_{n}) \quad \mbox{and} \quad	q_{n+2} > S_1(q_{n+1}),
	\end{eqnarray*}
	then the following holds for the unique lift $F:\tilde {\mathbb A}\rightarrow\tilde {\mathbb A}$ of $f$ with $\rho(F)\in(0,1)$:  let $\gamma$ be a $q_{n+1}$-good Brouwer curve
	for $f$ given by Lemma \ref{S1}\footnote{Lemma \ref{S1} is applicable since by hypothesis $S_2\geq S_1$.}, and let $\tilde\gamma$ be any lift of $\gamma$.    Let $\tilde          \mathcal D$ and $\tilde          \mathcal D_0$
	be the regions in $\tilde{\mathbb A}$ bounded by
	\begin{eqnarray*}
		\big( F^{q_{n},-p_{n}}(\tilde\gamma),\ F^{q_{n+1},-p_{n+1}}(\tilde\gamma) \big),\qquad
														\big(F^{q_{n},-p_{n}}(\tilde\gamma),\ \tilde\gamma\big)
	\end{eqnarray*}
	respectively.   Then there are open neighborhoods $\hat           \mathcal D$ of $\tilde           \mathcal D$ and $\hat           \mathcal D_0$ of $\tilde          \mathcal D_0$; a $C^\infty$-diffeomorphism
	\begin{eqnarray*}
			h : \hat          \mathcal D \to \hat          \mathcal D_0;
	\end{eqnarray*}
		and neighborhoods $U_L$ of $F^{q_{n}, -p_{n}}(\tilde\gamma)$ and
		$U_R$ of $F^{q_{n+1}, -p_{n+1}}(\tilde\gamma)$, so that
		\begin{eqnarray}
		&& h|_{U_L} = {\rm Id}, \quad h|_{U_R} = F^{-q_{n+1}, p_{n+1}},    \label{eq boundary}   \\
		&& \norm{h - {\rm Id}}_{\diff^{r_0}(\tilde           \mathcal D,\, \tilde          \mathcal D_0)} < Q_{r_0, M}\big(\beta_{n+1}, q_{n+1}\big).   \label{eq smooth bound for h}
		\end{eqnarray}
\end{lemma}
    \begin{proof}
	By Lemma \ref{S1} there is a diffeomorphism $\psi:\tilde{\mathbb A}\rightarrow\tilde{\mathbb A} $ satisfying
	(\ref{eq normofpsiyr2}) and which maps $[0,1]^2$ onto $\tilde          \mathcal D_0$.
	Set
	\begin{eqnarray*}
		\gamma' := F^{q_{n+1}, -p_{n+1}}(\tilde\gamma).
	\end{eqnarray*}
	By letting $q_{n+2}$ be sufficiently large depending on $q_{n+1}$ and $M$, we have $$ A_0(          \beta_{n+1}, M^{q_{n+1}}) < 1/2 $$
	where $A_0$ is produced by Theorem \ref{thm aflxz}.
	Then
	\begin{eqnarray*}
	 \| f^{q_{n+1}}- {\rm Id} \|
	\leq A_0(\|\rho(f^{q_{n+1}})\|_{{\mathbb R}/{\mathbb Z}}, M^{q_{n+1}})
	=  A_0(          \beta_{n+1}, M^{q_{n+1}}) < 1/2.
	\end{eqnarray*}
    Using Theorem \ref{thm aflxz}, Lemma \ref{lem:lift} and \eqref{eq continuefraction}, we obtain
    	\begin{eqnarray} \label{eq dhtildegamma'}
    	d_H(\tilde \gamma, \gamma')= d_H(f^{q_{n+1}}(\gamma),\gamma) &<&  A_0(          \beta_{n+1}, M^{q_{n+1}}).
    	\end{eqnarray}
    	Notice that by \eqref{eq normofpsiyr2} and \eqref{eq t psi 2} we also have that $\norm{\psi}_{\diff^{r_0}([0,2] \times [0,1])}$ is  bounded in terms of $q_{n+1}, M, r_{0}$.
	Then by \eqref{eq dhtildegamma'} and by letting $\beta_{n+1}$ be sufficiently large depending on $q_{n+1}, M, r_0$, we have
	\begin{eqnarray*}
	\gamma' \subset \psi([1/2, 3/2] \times [0,1]).
	\end{eqnarray*}
	We define
	\begin{eqnarray*}
	\gamma'' = \psi^{-1}(\gamma').
	\end{eqnarray*}
    	Then
	\begin{eqnarray*}
						\{1/2\} \times [0,1] < \gamma''< \left\{3/2\right\} \times [0,1]
	\end{eqnarray*}
	if $\beta_{n+1}$ is sufficiently large depending on $q_{n+1}, M, r_0$.
    	Denote by $\mathcal U'$ the region in $\tilde{\mathbb A}$ bounded by $\{0\}\times[0,1]$ and $\gamma''$.
    	It remains to construct a $C^\infty$-diffeomorphism
	$$\varphi: \mathcal U' \to [0,1]^2$$
	such that $\varphi$ equals ${\rm Id}$ near $\{0\} \times [0,1]$; equals $\psi^{-1} F^{-q_{n+1}, p_{n+1}} \psi$ near $\gamma''$;  and tends to ${\rm Id}$ in the $C^{r_0}$-topology as $\beta_{n+1}$ tends to $0$ for each fixed
	$q_{n+1}, M, r_0$.   Indeed, after the above $\varphi$ is constructed, we can define $h = \psi \varphi \psi^{-1}$.

    	By Theorem \ref{thm aflxz}, for  given $q_{n+1}, M, r_0$,
    	we see that $F^{-q_{n+1}, p_{n+1}}$ tends to  ${\rm Id}$ in the $C^{r_0}$-topology as $q_{n+2}$ tends to infinity.
    	Thus it suffices to construct $\varphi$ by smooth interpolation.
    \end{proof}

	Continuing the proof of Theorem \ref{thm. approximation}, we first construct a periodic approximation of $f$ as follows.  We define $G$ on $U_0$ (see \eqref{def u0}) by
	\begin{eqnarray} \label{eq def g}
	G|_{U} = \begin{cases}
		F &\quad  U = \cup_{i = 0}^{q_n - 2} {\rm int}( \tilde{          \mathcal B_i} ), \\
		F^{q_n} h F^{1-q_n} &\quad U = {\rm int}( {\tilde           \mathcal B}_{q_n-1}), \\
		F &\quad U =  \cup_{i = q_n}^{q_{n+1}-2} {\rm int}( {\tilde           \mathcal B}_i), \\
		T^{p_{n+1}}h^{-1} F^{1-q_{n+1}} &\quad U =  {\rm int}( {{\tilde           \mathcal B}}_{q_{n+1}-1}).
	\end{cases}
	\end{eqnarray}
	   \color{black}
	By property (1),(2) below \eqref{def u0}, we may extend $G$ to a self-map of $\tilde{\mathbb A} \setminus \Gamma$, still denoted by $G$, satisfying
	\begin{eqnarray} \label{def g t t g}
	G T = T G.
	\end{eqnarray}
	Moreover,
	by \eqref{eq def g} the following identity holds on $\tilde{\mathbb A} \setminus \Gamma$:
	\begin{eqnarray} \label{eq G T}
	G^{q_{n+1}} = T^{p_{n+1}}.
	\end{eqnarray}
	By construction, for any two curves $\gamma_a, \gamma_b$ in \eqref{def tgamma}, there are integers $i, j$ such that
	\begin{eqnarray} \label{eq gammaab}
	\gamma_b = T^i G^j(\gamma_a).
	\end{eqnarray}

By \eqref{eq def g} in the definition of $G$, we also have that
\begin{eqnarray} \label{b i eq g i b 0}
\tilde           \mathcal B_i = G^i(\tilde           \mathcal B_0), \quad 0 \leq i \leq q_{n+1}-1.
\end{eqnarray}
We denote
\begin{eqnarray} \label{eq bqn+1}
\tilde           \mathcal B_{q_{n+1}} := G^{q_{n+1}}(\tilde           \mathcal B_{0}) = T^{p_{n+1}}(\tilde           \mathcal B_{0}).
\end{eqnarray}

Now we set
\begin{eqnarray*}
\phi := \psi^{-1} h
\end{eqnarray*}
where $\psi$ is in Lemma \ref{S1} and $h$ is in Lemma \ref{S2}.
\begin{lemma} \label{lemma gqn}
	There is a neighborhood $V'\subset\tilde{\mathbb A}$ of $F^{q_{n+1}, -p_{n+1}}(\tilde \gamma)$ so that $G^{q_n}$ extends to a $C^\infty$ map on $V'$, and
\begin{eqnarray} \label{def phi}
T \phi(x) = \phi T^{-p_n} G^{q_n}(x), \quad x \in V'.
\end{eqnarray}
\end{lemma}
\begin{proof}
To see this, we first notice that
\begin{eqnarray*}
F^{q_{n+1}, -p_{n+1}}(\tilde \gamma) = \tilde           \mathcal B_{0} \cap T^{p_n - p_{n+1}}(\tilde           \mathcal B_{q_{n+1} - q_{n}}),
\end{eqnarray*}
and $ \tilde           \mathcal B_{0} \cup T^{p_n - p_{n+1}}(\tilde           \mathcal B_{q_{n+1} - q_{n}}) $ is a neighborhood of $F^{q_{n+1}, -p_{n+1}}(\tilde \gamma)$ in $\tilde{\mathbb A}$.
Let $U_L, U_R$ be given by Lemma \ref{S2}.
The set $V'$ defined by
\begin{eqnarray*}
V' = F^{q_{n+1} - q_n, -p_{n+1} + p_n}(U_L) \cap U_R \cap ( \tilde           \mathcal B_{0} \cup T^{p_n - p_{n+1}}(\tilde           \mathcal B_{q_{n+1} - q_{n}}) )
\end{eqnarray*}
is  a neighborhood of $F^{q_{n+1}, -p_{n+1}}(\tilde \gamma)$.

By \eqref{eq def g}, \eqref{b i eq g i b 0} and \eqref{def g t t g}, we have
\begin{eqnarray}
 G^{q_n}|_{Int(\tilde           \mathcal B_{0})} &=& F^{q_n} h, \label{eq gqn1} \\
 G^{q_n}|_{T^{p_n - p_{n+1}}(Int(\tilde           \mathcal B_{q_{n+1} - q_{n}}))} &=& T^{p_{n}}  h^{-1} F^{q_n - q_{n+1}}T^{p_{n+1} - p_{n}}. \label{eq gqn2}
\end{eqnarray}
By Lemma \ref{S2}, we see that
\begin{eqnarray*}
\mbox{RHS of \eqref{eq gqn1}}|_{V'}  = \mbox{RHS of \eqref{eq gqn2}}|_{V'}  =  T^{p_{n+1}}F^{q_n - q_{n+1}}.
\end{eqnarray*}
 Then it is clear that $G^{q_n}$ extends to a $C^\infty$ map on $V'$.
 Again by Lemma \ref{S2}, we have for any $x \in V'$
 \begin{eqnarray*}
 \phi T^{-p_n} G^{q_n}(x) &=& \psi^{-1} h T^{p_{n+1}-p_n} F^{q_n - q_{n+1}}(x) = \psi^{-1} T^{p_{n+1}-p_n} F^{q_n - q_{n+1}}(x), \\
 T \phi(x) &=& T \psi^{-1} h (x) =  T \psi^{-1} T^{-p_{n+1}} F^{q_{n+1}}(x).
 \end{eqnarray*}
Thus \eqref{def phi} follows from \eqref{eq t psi 2}.
\end{proof}

We have the following corollary.

\begin{cor} \label{lem defofG}
	The map $G$ on $\tilde{\mathbb A} \setminus \Gamma$ extends to an element in $\diff^\infty(\tilde{\mathbb A})$.
\end{cor}
\begin{proof}
By \eqref{eq G T}, clearly $G^{q_{n+1}}$ extends to a smooth map $T^{p_{n+1}}$ on $\tilde {\mathbb A}$. By Lemma \ref{lemma gqn}, \eqref{def g t t g}
 and  \eqref{eq gammaab}, we see that  $G^{q_{n}}$ also extends to a map in $\diff^\infty(\tilde {\mathbb A})$. To conclude the proof it suffices to notice that $$G = (G^{q_n})^{-p_{n+1}}(G^{q_{n+1}})^{p_n}.$$
\end{proof}

	By a slight abuse of notation, we again write $G\in \diff^\infty(\tilde{\mathbb A})$ for the extension provided in Corollary \ref{lem defofG}.   	Clearly \eqref{def g t t g} continues to hold.
	Consequently $G$ descends to a periodic diffeomorphism $g \in \diff^{\infty}({\mathbb A})$ satisfying
\begin{eqnarray*}
g^{q_{n+1}} = {\rm Id}.
\end{eqnarray*}

	We define map $\tilde H  : \tilde {\mathbb A} \to \tilde {\mathbb A}$ by
	\begin{align}
	\qquad \tilde H(x) := T^{-kq_{n+1} + jp_{n+1}}\phi G^{-j}T^{k }(x), \quad x \in T^{-k}\tilde           \mathcal B_j \label{def tilde H}
	\end{align}
	for all $ k \in {\mathbb Z}$ and all $0 \leq j \leq q_{n+1}-1$.
	Then by \eqref{eq G T}, \eqref{b i eq g i b 0}, \eqref{eq bqn+1} and \eqref{def tilde H} we have
	\begin{eqnarray}\label{tilde H is equivariant}
	\tilde H T = T^{q_{n+1}} \tilde H, \quad	\tilde H G = T^{p_{n+1}} \tilde H.
	\end{eqnarray}
	We set $H_1 := D_{q_{n+1}} \tilde H$, where
	\begin{eqnarray*}
	D_{\lambda}(x,y) = (\lambda^{-1}x, y), \quad \lambda \in {\mathbb R} \setminus \{0\}.
	\end{eqnarray*}
	Notice that for every $\lambda \in {\mathbb R} \setminus \{0\}$ and every $p \in {\mathbb R}$ we have
	\begin{eqnarray*}
	D_{\lambda} T_p D_{\lambda}^{-1} = T_{\frac{p}{\lambda}}
	\end{eqnarray*}
	where for each $c \in {\mathbb R}$, $T_c:\tilde{\mathbb A}\to \tilde{\mathbb A}$ denotes the map $T_c(x,y) = (x+c,y)$.
	Then by \eqref{tilde H is equivariant}, we see that $H_1:\tilde{\mathbb A}\to \tilde{\mathbb A}$ satisfies
	\begin{eqnarray} \label{eq h1descends}
	TH_1 = H_1T, \quad H_1 G = T_{p_{n+1}/q_{n+1}}H_1.
	\end{eqnarray}

	We now show that $\tilde H \in \diff^\infty(\tilde{\mathbb A})$. By construction, $\tilde H$ is $C^\infty$-smooth in the interior of each $T^{-k} \tilde B_j$, $k \in {\mathbb Z}$, $0 \leq j \leq q_{n+1}-1$. Thus it remains to show that $\tilde H$ is $C^\infty$ in a neighborhood of each of the curves in \eqref{def tgamma}.

	By \eqref{eq defoftildeb}, we can see that the set    \color{black}
	\begin{eqnarray}
    \tilde           \mathcal B_* :=  \tilde           \mathcal B_0 \cup T^{-p_n}\tilde           \mathcal B_{q_n}
	\end{eqnarray}
	is a neighborhood of $T^{-p_n} F^{q_n}(\tilde\gamma)$.
	By \eqref{eq gammaab}, we deduce that every curve in \eqref{def tgamma} has a neighborhood of the form $T^iG^j(\tilde           \mathcal B_*)$. By \eqref{tilde H is equivariant}, it remains to verify that 	$\tilde H$ is $C^\infty$ over  $\tilde           \mathcal B_*$.
	This follows from a similar argument as in the proof of Lemma \ref{lemma gqn} using \eqref{def phi} and \eqref{def tilde H}.
	Consequently $\tilde H$ and $H_1$ are $C^\infty$-smooth diffeomorphisms of $\tilde{\mathbb A}$.
	Then by \eqref{eq h1descends} $H_1$ descends to a $C^{\infty}$-diffeomorphism $h_1: {\mathbb A} \to {\mathbb A}$ satisfying
	\begin{eqnarray} \label{def g}
	 g = h_1^{-1} R_{p_{n+1}/q_{n+1}} h_1.
	\end{eqnarray}
	Moreover, we have the claim that:

	\smallskip
	 {\it $\|H_1\|_{\diff^{r_0}(\tilde{\mathbb A})}$, and hence $\|h_1\|_{\diff^{r_0}({\mathbb A})}$ as well, are bounded in terms of $q_{n+1}, M, r_0$. }

	Indeed, by definition it is clear that $\norm{\phi}_{\diff^{r_0}}$ is bounded in terms of $q_{n+1}, M, r_0$; and for each integer $0 \leq j \leq q_{n+1}-1$, $\norm{G}_{\diff^{r_0}(\tilde          \mathcal B_{j}, \tilde          \mathcal B_{j+1})}$ is bounded in terms of $q_{n+1}, M, r_0$. Our claim follows immediately from \eqref{def tilde H} and the fact that $H_1$ commutes with $T$.

	Recall that $r_0$, $M$ and $\epsilon$ are fixed.
	In the following we will show that:
	\begin{lemma}\label{lem h2}
		For sufficiently fast growing $P$ depending on $r_0, M$ and $\epsilon$ the following holds. Assume $n$ satisfies the conditions in Lemma \ref{S1} and Lemma \ref{S2}; and moreover \eqref{eq qn+2qn+1qnP} holds for $n$. Then we have:
	\begin{enumerate}
	\item $d_{\diff^{r_0}({\mathbb A})}(f,g) < \epsilon/4$;
	\item $d_{\diff^{r_0}({\mathbb A})}(h_1^{-1} R_{{\alpha}} h_1, g) < \epsilon/4$;
	\item there is $h_2 \in \diff^\infty( {\mathbb A} )$ such that
	$(h_2h_1)_* \omega = \omega$ and
	\begin{eqnarray*}
	d_{\diff^{r_0 - 1}({\mathbb A})}(h_1^{-1}h_2^{-1} R_{{\alpha}} h_2h_1, h_1^{-1} R_{{\alpha}} h_1) < \epsilon/4.
	\end{eqnarray*}
	\end{enumerate}
\end{lemma}
\begin{proof}
	\noindent{\it Proof of $(1)$}:
	By \eqref{eq def g}, there is $D > 0$ depending only on $r_0$ such that
	\begin{eqnarray*}
	 \quad d_{\diff^{r_0}({\mathbb A})}(f,g) \leq D  \max(d_{\diff^{r_0}({\mathbb A})}(f^{q_{n}} h f^{-q_{n}},{\rm Id}), d_{\diff^{r_0}({\mathbb A})}( h^{-1} f^{-q_{n+1}},{\rm Id})).\nonumber
	\end{eqnarray*}
	By Theorem \ref{thm aflxz}, Lemma \ref{S2}, and by letting $q_{n+2}$ be sufficiently large depending on $\epsilon, r_0, M, q_{n+1}$, we have
	\begin{eqnarray*}
	d_{\diff^{r_0}({\mathbb A})}(f^{q_{n}} h f^{-q_{n}},{\rm Id}), \quad
	 d_{\diff^{r_0}({\mathbb A})}( h^{-1} f^{-q_{n+1}},{\rm Id})  < (4D)^{-1}\epsilon.
	\end{eqnarray*}
	Thus we have
	\begin{eqnarray*}
		d_{\diff^{r_0}({\mathbb A})}(f,g) < \epsilon/4.
	\end{eqnarray*}

	\noindent{\it Proof of $(2)$}:
		Since $\norm{h_1}_{\diff^{r_0}({\mathbb A})}$ admits an upper bound depending only on $q_{n+1}, M$ and $r_0$,  by letting $q_{n+2}$ be sufficiently large depending on $\epsilon, r_0, M, q_{n+1}$, we have
	\begin{eqnarray*}
	d_{\diff^{r_0}({\mathbb A})}(h_1^{-1} R_{{\alpha}} h_1,g) <  \epsilon/4.
	\end{eqnarray*}

	\noindent{\it Proof of $(3)$}:
	Set
	\begin{eqnarray*}
	\lambda_0 = \det(H_1) = \frac{1}{q_{n+1}} \det(\tilde H)  \in C^\infty(\tilde {\mathbb A}).
	\end{eqnarray*}
	Recall that both $\psi$ and $F$ have constant Jacobians, hence
	by \eqref{def tilde H}, on each $\tilde          \mathcal B_j$  we have
	\begin{eqnarray*}
	\log \lambda_0 =  \log  \det h \circ (G^{-j}) + \log \det (G^{-j} F^j)\circ F^j + c_1
	\end{eqnarray*}
	for some constant $c_1$. On $T^{-k}\tilde           \mathcal B_j$ we have a similar formula.
	Thus
	\begin{eqnarray*}
	\sup_{1 \leq r \leq r_0}\norm{D^r\log\lambda_0}  \leq (1 + d_{\diff^{r_0}({\mathbb A})}(f, {\rm Id}))^{ C q_{n+1} r_0}( d_{\diff^{r_0}({\mathbb A})}(h, {\rm Id}) + d_{\diff^{r_0}({\mathbb A})}(g, f)).
	\end{eqnarray*}
By Lemma \ref{S2} and the argument above for bounding  $d_{\diff^{r_0}({\mathbb A})}(g, f)$, we see that
$\sup_{1 \leq r \leq r_0}\norm{D^r\log\lambda_0}$ can be made arbitrarily small by making $q_{n+2}$ sufficiently large while keeping $q_{n+1}, M$ and $r_0$ fixed. By \eqref{eq h1descends}, we have
\begin{eqnarray*}
\int_{[0,1]^2} \lambda_0 d\omega = 1.
\end{eqnarray*}
Then it is direct to see, for some absolute constant $C > 0$, that
\begin{eqnarray*}
\norm{\lambda_0 -1 } < C \norm{D \log \lambda_0}
\end{eqnarray*}
given that $\norm{D\log \lambda_0} < 1$.
Consequently, for any $\delta > 0$, we have $\norm{\lambda_0 -1}_{C^{r_0}} < \delta$ if $q_{n+2}$ is sufficiently large depending on $\delta, q_{n+1}, M$ and $r_0$.

    By Dacorogna-Moser's theorem  (\cite[Theorem 1]{DacMos}), there exists $h_2 \in \diff^\infty( {\mathbb A} )$ such that
    	\begin{eqnarray*}
    	(h_2)_* (\lambda_0\omega) = \omega.
    	\end{eqnarray*}
    Moreover, by \cite[Theorem 2 and Lemma 3]{DacMos}, we can choose $h_2$ with $d_{\diff^{r_0 - 1}({\mathbb A})}(h_2, {\rm Id})$ arbitrarily small provided $\norm{\lambda_0 -1}_{C^{r_0}}$ is sufficiently small.

    In summary, if $q_{n+2}$ is sufficiently large depending on $\epsilon, q_{n+1}, M$ and $r_0$, we can choose $h_2 \in \diff^{\infty}({\mathbb A})$ sufficiently close to ${\rm Id}$ in $\diff^{r_0}({\mathbb A})$ so that
    \begin{eqnarray*}
    d_{\diff^{r_0 - 1}({\mathbb A})}(h_1^{-1}h_2^{-1} R_{{\alpha}} h_2h_1, h_1^{-1} R_{{\alpha}} h_1) < \epsilon/4.
    \end{eqnarray*}
\end{proof}
     We set $h_0 = h_2h_1$. By Lemma \ref{lem h2}(3) we have $h_0 \in \diff^{\infty}({\mathbb A}, \omega)$.
	Then by Lemma \ref{lem h2}, we have
	\begin{eqnarray*}
	d_{\diff^{r_0	}({\mathbb A})}(h_0^{-1} R_{{\alpha}} h_0,f) <  \epsilon
	\end{eqnarray*}
	This completes the proof of Theorem \ref{thm. approximation}.
\end{proof}

We can now prove the main result with the aid of the following lemma whose straightforward proof is omitted.

\begin{lemma}\label{lem construct rotation nbs}
		For any  function $P: {\mathbb N} \to {\mathbb N}$, the set
		\begin{eqnarray*}
		\ \           \mathcal C := \{ \alpha \in (0,1)\setminus {\mathbb Q} &\mid& \exists n\geq 3 \mbox{ odd such that } q_{n}(\alpha) > P(q_{n-1}({\alpha})), \\
		&&q_{n+1}(\alpha) > P(q_{n}({\alpha})),  q_{n+2}(\alpha) > P(q_{n+1}({\alpha}))  \}
		\end{eqnarray*}
		is open and dense in $(0,1)\setminus {\mathbb Q}$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{main thm F = closure O}]
We have seen in the introduction that $\overline{O^{\infty}_{{\mathbb A}}({\alpha})}\subset F^{\infty}_{{\mathbb A}}({\alpha})$ for any ${\alpha} \in (0,1)/{\mathbb Q}$. It remains to show that for a Baire generic ${\alpha} \in (0,1)/{\mathbb Q}$
	\begin{eqnarray} \label{eq finoclosure}
	F^{\infty}_{\mathbb A}({\alpha}) \subset \overline{O^\infty_{\mathbb A}({\alpha})}
	\end{eqnarray}
with the closure taken in the $C^\infty$-topology.   For each tuple $(r_0,M,k)$ in ${\mathbb Z}_{\geq 1}^3$, with $r_0\geq 2$, let
$P_{r_0,M,k^{-1}}$ be the function produced by Theorem \ref{thm. approximation}.    Therefore by Lemma \ref{lem construct rotation nbs} the set
\begin{align*}
		&	          \mathcal A(r_0,M,k):= \Big\{  \alpha \in (0,1)\setminus {\mathbb Q} \mid   \mbox{there is an odd } n\geq 3 \mbox{ such that }\\
								&q_{n}(\alpha) >  P_{r_0,M,k^{-1}}(q_{n-1}(\alpha)), \quad	q_{n+1}(\alpha) > P_{r_0,M,k^{-1}}(q_{n}(\alpha)) \\
 \mbox{ and}&\quad q_{n+2}(\alpha) > P_{r_0,M,k^{-1}}(q_{n+1}(\alpha))  \Big\}
\end{align*}
is open and dense in $(0,1)\setminus {\mathbb Q}$.    Thus the countable intersection
	\begin{eqnarray*}
	          \mathcal A := \cap_{r \geq 2} \cap_{k \geq 1} \cap_{M \geq 2}           \mathcal A(r, M, k)
	\end{eqnarray*}
	is a residual subset of $(0,1)\setminus {\mathbb Q}$.  Fix ${\alpha}\in          \mathcal A$ and
	suppose $f \in F^\infty_{{\mathbb A}}({\alpha})$.   Then for any $\epsilon > 0$ and $r\in{\mathbb N}$ with $r\geq 2$ choose $M\in{\mathbb N}$ so that $\norm{f}_{\diff^{r+2}({\mathbb A})} < M$,
	then by ${\alpha} \in           \mathcal A(r, M, \lceil \epsilon \rceil^{-1})$ and by Theorem \ref{thm. approximation} there exists $h \in \diff^{\infty}({\mathbb A}, \omega)$ so that
	\begin{eqnarray*}
	d_{C^{r - 1}}(f, hR_{\alpha} h^{-1}) < \epsilon.
	\end{eqnarray*}
	This gives $F^{\infty}_{\mathbb A}({\alpha}) \subset \overline{O^\infty_{\mathbb A}({\alpha})}$ with the closure in the $C^{r- 1}$-topology.   Since $r$ is arbitrary we easily conclude \eqref{eq finoclosure} holds with closure in the $C^\infty$-topology.
\end{proof}

\begin{comment}

	\begin{proof}
	Let $\mathcal U_0$ be an open neighborhood of $\gamma$ in ${\mathbb A}$ such that
	\begin{eqnarray*}
	g(\mathcal U_0) \cap \mathcal U_0 = \emptyset.
	\end{eqnarray*}

	We claim that there is a $C^\infty$-smooth Brouwer curve $\gamma$ contained in $\mathcal U_0$,
	and an orientation-preserving $C^r$-smooth $\psi_L: \mathcal U_L \to {\mathbb A}$ which is a diffeomorphism from an open neighborhood $\mathcal U_L$ of
	$\{0\} \times [0,1]$ onto its image in ${\mathbb A}$, so that
	\begin{eqnarray*}
	\det (D\psi_L) \equiv \omega(\mathcal R) \quad \mbox{ and } \quad
	\psi_L(\{0\} \times [0,1]) = \gamma.
	\end{eqnarray*}
	Here $\mathcal R\subset{\mathbb A}$ denotes the region bounded by $(\gamma, g(\gamma))$, and $\omega(\mathcal R)$ denotes its $\omega$-volume.
   Indeed, we can first construct $\psi_L$ to have constant Jacobian near the two end points of $\{0\}\times[0,1]$, then composing with a
	further diffeomorphism supplied by Dacoragna-Moser \cite{DacMos} we can assume that the Jacobian of
	$\psi_L$ is constant on $\mathcal U_L$.

	Now set $\mathcal U_R = T (\mathcal U_L)$ and define $\psi_R: \mathcal U_R \to {\mathbb A}$ by
	\begin{eqnarray*}
	\psi_R = g \psi_L T^{-1}.
	\end{eqnarray*}
	By taking $\mathcal U_L$ sufficiently small, we may assume that
	\begin{eqnarray*}
	\psi_L(\mathcal U_L) \cap \psi_R(\mathcal U_R) = \emptyset.
	\end{eqnarray*}
	Thus $\psi_L$ and $\psi_R$ together define a map on the union $\mathcal U:=\mathcal U_L\cup\mathcal U_R$
	\begin{eqnarray*}
							\psi_0:\mathcal U\rightarrow{\mathbb A}
	\end{eqnarray*}
	which is a $C^r$-diffeomorphism onto its image and which pulls back $\omega$ to $\omega_0|_{\mathcal U}$ where we denote
	\begin{eqnarray*}
	\omega_0 := \omega(\mathcal R)\omega
	\end{eqnarray*}
	on $\tilde{\mathbb A}$.   Moreover, by construction the $\omega$-volume of $[0,1]^2 \setminus \mathcal U$ equals the $\omega_0$-volume of
	$\mathcal R \setminus \psi(\mathcal U)$.

			We fix a small constant $\sigma > 0$ and denote
		\begin{eqnarray*}
		\mathcal V^{(i)}_L = [-i\sigma, i\sigma] \times [0,1], \quad i = 1,2,3.
		\end{eqnarray*}
		Moreover, we require that
		\begin{eqnarray*}
		\mathcal V^{(3)}_L \subset \mathcal U_L.
		\end{eqnarray*}
		We denote $\mathcal V_R^{(i)} = T(\mathcal V_L^{(i)})$ and $\mathcal V^{(i)}  = \mathcal V_L^{(i)}  \cup \mathcal V_R^{(i)}$ for $i=1,2,3$.

	We first construct a map $\psi_1 : [0,1]^2 \cup \mathcal U \to {\mathbb A}$ that is a $C^r$-diffeomorphism onto its image, and coincides with $\psi_0$ on $\mathcal V^{(2)}$.  Moreover it is clear that for any $r \geq 1$ we can make the $C^r$-norm of $\psi_1$ depend only on the $C^r$-norm of $\psi_0$.
	Denote
	\begin{eqnarray} \label{eq pullback}
	\omega_1 = \psi_1^{*}\omega.
	\end{eqnarray}
	Then there is a positive $C^\infty$-function $\varrho: [0,1]^2 \cup \mathcal V^{(2)} \to {\mathbb R}$ such that
	\begin{eqnarray*}
	\omega_1 = \varrho \omega_0.
	\end{eqnarray*}
	Moreover, since $\psi_1$ concides with $\psi_0$ on $\mathcal V^{(2)}$, we see that
	\begin{eqnarray*}
	\varrho|_{\mathcal V^{(2)}} \equiv 1.
	\end{eqnarray*}
	We have the following lemma.
	\begin{lemma}\label{lem modifyvolume}
		There is a $C^\infty$-diffeomorphism $\phi: [0,1]^2 \cup \mathcal V^{(1)} \to [0,1]^2 \cup \mathcal V^{(1)} $ which equals the identity on $\mathcal V^{(1)}$, and satisfies that
		\begin{eqnarray} \label{eq pushforward}
		\phi_* \omega_0 = \omega_1.
		\end{eqnarray}
	\end{lemma}
\begin{proof}
	The proof is essentially contained in \cite[Corollary 5]{A10}, which is based on \cite{DacMos}. We give a brief outline of the proof for the convenience of the readers.

	We may suppose that $\varrho$ is $C^\infty$ close to $1$. Indeed, once we have verified the lemma for such $\varrho$, we can prove the statement for a general $\varrho$ by concatenation.

	By \cite[Theorem 2]{DacMos}, we can find a $C^\infty$ map $w: [0,1]^2 \to {\mathbb R}^2$ with $div(w) = \varrho - 1$ and $w|_{\partial [0,1]^2} = 0$, and if $\varrho-1$ is $C^\infty$ small then $w$ is $C^\infty$ small as well.
	Now the proof of \cite[Theorem 2]{A10} implies that there is a $C^\infty$ map $u: [0,1]^2 \to {\mathbb R}^2$ such that $div(u) = 0$ and $u|_{\mathcal V^{(1)} \cap [0,1]^2} = w$.  Then the map $v: [0,1]^2 \to {\mathbb R}^2$ defined by $v = w-u$ satisfies that $div(v) = \varrho - 1 $ and $v|_{\mathcal V^{(1)} \cap [0,1]^2} = 0$.

	By Step I in the proof of \cite[Lemma 3]{DacMos} (see also \cite[Theorem 3]{A10}), we see that $\phi = \phi_1$ verifies \eqref{eq pushforward} where $\phi_t(x)$ is  the solution of the differential equation $\frac{d}{dt}\phi_t(x) = v(\phi_t(x))/(t + (1-t)(1 + (\varrho - 1)(\phi_t(x))))$ with $\phi_0(x) = x$. In particular, we see that $\phi_t(x) = x$ for any $x \in \mathcal V^{(1)} \cap [0,1]^2$ and $t \in [0,1]$. This concludes the proof.
\end{proof}
We define
\begin{eqnarray*}
\psi = \psi_1 \phi.
\end{eqnarray*}
By \eqref{eq pullback} and Lemma \ref{lem modifyvolume}, we see that
\begin{eqnarray*}
\psi_* \omega_0 = \omega \ \mbox{ and } \ \psi|_{\mathcal V^{(1)}} = \psi_0|_{\mathcal V^{(1)}}.
\end{eqnarray*}
Thus $\psi$ gives rise to  a $C^r$-admissible coordinate for $(\mathcal R, g)$.   The above construction applies to each $g'$ that is sufficiently close to $g$ in the $C^r$-topology, so
that the resulting $C^r$-admissible coordinates for $g'$ are $C^r$-close to those of $g$.
\end{proof}

\end{comment}

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