text:
\section{Introduction}
Throughout $G_0$ denotes a (connected) semi-simple Lie group which is
acting on a flag manifold $Z=G/Q$ of its complexification $G$.  If $G_0$ is
not simple, then all objects considered here split into products according to
a product decomposition of $G_0$ and therefore we assume that $G_0$ is
simple.  Since $G_0$ has only finitely many orbits on $Z$ (\cite{W}), it has open
orbits $D$ which are referred to as \emph{flag domains}.   If $G_0$ is a compact
real form of $G$, then $D=Z$, a situation that is not of interest from the point of
view of this paper.  There are also exceptional situations where $G_0$ is a non-compact
real form, but nevertheless $D=Z$ (see, e.g., \cite{FHW} for this and other background).
We also assume there that this is also not the case, i.e., we only consider flag domains  $D$
which are proper open subsets of $Z$.  Here we focus on certain complex geometric properties
of these domains which involve basic compact complex submanifolds which are associated
to the group actions at hand.  We begin with background information followed by a statement
of the first version of our results (Theorem \ref{weak pseudoconcavity}).

\subsection {Convexity and concavity}
\subsubsection {Generalities}
A complex space $X$ is a Stein space if and only if the holomorphic functions on $X$ separate its
points, and given any divergent sequence $\{x_n\}$ in $X$ there exists a holomorphic function
$f\in \mathcal{O}(X)$ with $\lim \vert f(x_n)\vert =\infty $.  If only the second condition is fulfilled, then
$X$ is said to be holomorphically convex.  In this case there is a canonical proper, surjective holomorphic
map $X\to X_{red}$ onto a Stein space.  At the level of sets this \emph{Remmert reduction} is defined
by the equivalence relation $x\sim y$ whenever $f(x)=f(y)$ for all $f\in \mathcal{O}(X)$. If $\mathcal{S}$
is a coherent sheaf on $X$, or more simply a holomorphic vector bundle, and $X$ is Stein, then
all cohomology groups $H^q(X,\mathcal{S})$ vanish for $q\ge 1$.  If $X$ is holomorphically convex,
then in principle these groups can be computed from the Remmert reduction; in particular, they are
finite-dimensional.  Since many problems in complex analysis can be stated in terms of cohomological
invariants, these results are of fundamental importance (see, e.g., \cite{GR1}, \cite{GR2}).

\bigskip\noindent
The role of convexity in Stein theory can be exemplified as follows: Suppose $\Omega $ is a relatively
compact domain with smooth boundary in a complex manifold $X$, e.g., $X=\mathbb{C}^n$.   Let us suppose
that every point $p\in \partial \Omega $ possesses a coordinate ball $B_p$ in $X$ in which there
is a $1$-codimensional complex submanifold $H_p$ which contains $p$ but which otherwise is contained
in the complement of the closure of $\Omega $.  In this setting there exist holomorphic functions $h\in \mathcal{O}(B_p)$
which vanish exactly on $H_p$.  Thus, if $\{x_n\}$ is a sequence in $B_p\cap \Omega $ which converges to $p$ and $h$ is such a function,
then $f:=\frac{1}{h}$ is such that $lim \vert f(x_n)\vert =\infty $.  In this situation one says $\Omega $ has a supporting
holomorphic supporting hypersurfaces at each of its boundary points, a condition that is reminiscent of classical convexity.
Under further natural conditions on $X$, e.g., if $X$ itself is Stein, one can in fact show that if $\Omega $ is convex in this
strong sense, then it is Stein.

\bigskip\noindent
It is convenient to discuss supporting manifolds such as the hypersurfaces $H_p$ in terms of local holomorphic maps
of $d$-dimensional polydisks $\Delta =\Delta_d(r):=\{z\in \mathbb{C}^d: \vert z_i\vert \le r \ \text{for all}\  i\}$.  A $d$-dimensional
polydisk through $p$ in a complex manifold $X$ is a map $F:\Delta \to X$ which is holomorphic with $rank_z(dF)=d$
at all points $z\in \Delta $ and with $F(0)=p$.  On the convexity side we can localize the hypersurfaces $H_p$ discussed
above so that they are parameterized by such maps.   Andreotti defined the notion of \emph{pseudoconcavity} analogously.
The following definition is a bit stronger than Andreotti's (\cite{A}), but is appropriate for our present considerations.
\begin {definition}
A connected complex manifold $X$ is said to be \emph{pseudoconcave of degree $d$} if it contains a
relatively compact open set $Y$ so that for every $p\in \partial Y$ there exists a holomorphic map $F:\Delta_d(r)\to X$
through $p$ with boundary $F(\partial \Delta)$ contained in $Y$.
\end {definition}
\noindent
Andreotti's first theorem in this setting is that if $X$ is \emph{pseudoconcave}, i.e., of degree $d=1$, then
$H^0(X,\mathcal{S})$ is finite-dimensional for all coherent sheaves $\mathcal{S}$, i.e. finiteness of cohomology
at level $0$. If $X$ can be smoothly exhausted by $\rho :X\to \mathbb{R} ^{\ge 0}$ whose sublevel sets
$Y_r:=\{\rho <r\}$ are, for $r$ sufficiently large,  $d$-pseudoconcave, then finiteness of cohomology holds
for $H^q(X,\mathcal{S})$ for $q\le d-1$ (\cite{AG}).

\subsubsection {Convexity and concavity of flag domains.}
Every Hermitian symmetric space of non-compact type (HSS) is an example of a flag domain $D_{SS}$.  Such
is canonically realized as an orbit of a real form $G_0$ (of Hermitian type) in a flag manifold $Z_{SS}$ which is the
associated Hermitian symmetric  space of compact type.   Using this embedding of $D_{SS}$ one
realizes it as a bounded Stein domain in the orbit of an associated unipotent
group which is a copy of $\mathbb{C}^n$.   More generally, if $D$ is any $G_0$-flag domain in a
$G$-flag manifold $Z$ and $\mathcal{O}(D)\not=\mathbb{C}$, then $G_0$ is of Hermitian
type, $D$ is holomorphically convex and the Remmert reduction is given as the
(saturated) restriction of a $G$-equivariant projection $Z\to Z_{SS}$ to a $G_0$-fibration
$D\to D_{SS}$.   This situation is discussed in more detail below, but here let us
just note that the fiber $C$ of this bundle is a compact submanifold of $D$ which is canonically
associated to the group actions at hand. Since $D_{SS}$ is Stein and contractible,
the bundle is holomorphically trivial, i.e., $D$ is bundle equivalent to $D_{SS}\times C$.
Needless to say, the holomorphically convex flag domains are of a very special type.

\bigskip\noindent
The following is a first (qualitative) version of the main result of this work.
\begin {theorem}\label{weak pseudoconcavity}
A flag domain is pseudoconcave if and only if it is not holomorphically convex.
\end {theorem}
\noindent
The sets $Y$, which display the pseudoconcavity of $D$, are constructed in such a way
that in many situations  the dimensions of the  supporting polydisks can be explicitly computed
in terms of certain root theoretic invariants (see Corollary \ref{pseudoconcave}).  This leads to refined estimates
on the degree $d$ of the pseudoconcavity.  Since the sets $Y$ can be realized as sublevel sets
of (not necessarily smooth) exhaustions,  one would hope that finiteness theorems for higher
degree cohomology can be proved.

\bigskip\noindent
In ([Hu]) it was shown that a flag domain $D$ possesses non-constant holomorphic functions if and only if it is pseudoconvex.  This condition
is also equivalent to $D$ being holomorphic convex.  It was conjectured there that otherwise $D$ should be pseudoconcave.  With this in mind
we have the following reformulation of Theorem \ref{weak pseudoconcavity}.
\
\begin {corollary}
A necessary and sufficient condition for a flag domain to be pseudoconcave is that it is not pseudoconvex.
\end {corollary}

\section {Background on cycles and cycle spaces}
The supporting polydisks for the realization of the pseudoconcavity of a given flag domain $D$
are contained in important compact submanifolds of $D$ which are defined by the group actions
at hand.  The starting point for the discussion is the fact that every maximal compact subgroup $K_0$ of $G_0$
has a unique orbit $C_0$ in $D$ which is a complex submanifold.  Such  \emph{base cycles} can also
be characterized as the unique compact orbits in $D$ of the complexification $K$ of $K_0$.   A special
case of Matsuki duality (\cite{W}) states that there is a $1-1$ correspondence between the  closed
$K$-orbits $C_0$ in $Z$ and the flag domains which contain them.

\bigskip\noindent
Letting $q:=dim_{\mathbb{C}} C_0$, we view $C_0$ as being moved by the algebraic $G$-action on the
space $\mathcal{C}_q(Z)$ of $q$-dimensional cycles in $Z$.  For our purposes here it is enough to
consider the orbit $\mathcal{M}_Z:=G.[C_0]$ and its open subset $\mathcal{M}_D$ which is defined to
be the connected component containing $[C_0]$ of the set of cycles in $\mathcal{M}_Z$ which are contained
in $D$.  Clearly $K$ is contained in the isotropy group $G_{[C_0]}$, but this isotropy group can be larger:
If $Q=G_p$ is the isotropy group at a point $p\in C_0$, it is possible that $Q$ also stabilizes $C_0$ and
that the parabolic group $P:=QK$ is $G_{[C_0]}$ (see \cite{W} and \cite{FHW} for this and other related background
material).\\

\noindent
\textbf{Remark}\label{isotropy}
Below we describe the two ways that this can occur.  Both are in the case where $G_0$ is of Hermitian type,
i.e., where the center of $K_0$ is $1$-dimensional.  As background,  let us comment that In both the Hermitian and non-Hermitian cases $K_0$
is a maximal subgroup of $G_0$ (see \cite{Hu2} for a proof).  In the non-Hermitian case the complex Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{s}$
is such that the $\mathfrak{k}$-representation on $\mathfrak{s}$ is irreducible. Therefore the connected complex group $K$ is maximal in the
sense that no proper Lie subgroup of $G$ contains it properly.  Since $K$ is algebraic, its normalizer is likewise algebraic and consequently in the non-Hermitian case
the only proper subgroups which contain $K$ are finite extensions of it.   In the Hermitian case $\mathfrak{s}=\mathfrak{s}^-\oplus \mathfrak{s}^+$
splits into two irreducible $\mathfrak{k}$-modules with $\mathfrak{p}^-=\mathfrak{k}\oplus \mathfrak{s}^- $ and $\mathfrak{p}^+=\mathfrak{k}\oplus \mathfrak{s}^+$
being parabolic. Other than $G$, the associated parabolic subgroups $P^-$ and $P^+$ are the only Lie subgroups which properly contain $K$. In this case, finite extensions
lying between $K$ and such a parabolic $P$ are trivial, because $P/K$ is isomorphic to $\mathbb{C}^n$ which does not support a fixed point free action of
a finite group (see \cite{W} and \cite{FHW} for this and other related background material).\qed

\bigskip\noindent
\begin{example}
Recall that if $G_0$ is of Hermitian type and $D_{SS}$ is one of its Hermitian
symmetric spaces of non-compact type,  then $D_{SS}$ is a $G_0$-flag domain in its compact
dual $Z_{SS}$.  Since $D_{SS}$ is Stein, $C_0$ consists of just one point $p_0$ and therefore
$G_{[C_0]}=G_{p_0}=:P$ with $Z_{SS}=G/P$ and $K\subset P$. Now suppose that $Z$ is another flag manifold with
$G$-equivariant projection $\pi :Z\to Z_{SS}$ and let $D$ be a flag domain with $\pi(D)=D_{SS}$.  For the
same reason as above, the base cycle $C_0$ is mapped to a point $p_0$ with $G_{p_0}=:P$ containing
$K$. It also contains the isotropy subgroup $Q=G_{q_0}$ of any point $q_0\in C_0$, because for every $g\in Q$ Remmert's
proper mapping theorem implies that
$\pi (g(C_0))$ is a connected compact subvariety containing $p_0$ in $D_{SS}$.  Since $D_{SS}$ is holomorphically
separable, it follows that $\pi (g(C_0))=\{p_0\}$. Thus it follows that
that $P$ is either $P^+$ or $P^-$ with $K.Q=P$. In other words
$D$ is the full $\pi $-preimage of $D_{SS}$ and the $C_0$ is just the fiber over the base point $p_0$.
In this case $\mathcal{M}_D=D_{SS}$ and $\mathcal{M}_Z=Z_{SS}$.  In this special situation we will
say that the flag domain $D$ \emph{fibers over a HSS}.
\end {example}

\bigskip\noindent
There is one other special situation where the isotropy $G_{[C_0]}$ is essentially larger than $K$.
 Here is a concrete example of how this occurs.

 \bigskip\noindent
 \begin {example}

 Let $G_0:=SU(1,n)$ act on $Z=\mathbb{P}(\mathbb{C}^{n+1})$ as usual.   It has
 two open orbits, the set of negative $1$-dimensional subspaces, which we denote by $D_-$
 and which is biholomorphically equivalent to the unit Euclidean ball in $\mathbb{C}^n$, and
 the set $D_+$ of positive subspaces, i.e., the complement of the closure of $D_-$.  Let
 $E_-:=Span\{e_1\}$, $E_+:=Span\{e_2,\ldots ,e_{n+1}\}$ and $K_0$ be the stabilizer in $G_0$ of the
 decomposition $\mathbb{C}^{n+1}=E_-\oplus E_+$.  The base cycle in $D_-$ is just the $K_0$-fixed
 point $\mathbb{P}(E_-)$ and the base cycle in $D_+$ is the projective hyperplane $C_0:=\mathbb{P}(E_+)$.
 The Borel subgroup $B$ of upper triangular matrices stabilizes $C_0$ so that $P:=BK$ is its full stabilizer.
 (This is of course just the isotropy group of the associated point in the dual projective space.)
 \end {example}

 \bigskip\noindent
 The following is the general setting of the above example.

 \bigskip\noindent
 \textit{Example 2 (general).}  Assume that $G_0$ is of Hermitian type.  Suppose that $D$ is a flag domain in $Z$ with base cycle $C_0$ and let $p_0\in C_0$.
 Let $\pi :\hat{Z}\to Z$ be the projection from the manifold of full flags and $\hat{D}$ be a flag domain in
 $\hat{Z}$ which holomorphically maps to $D_{SS}$ (see Example 1.).  Assume that $\pi (\hat{D})=D$.  Now let $B$ be the Borel subgroup of $G$ defined by a point
 $q_0$ in the base cycle $\hat{C}_0$ with $\pi (q_0)=p_0$.  Since $\hat{D}$ maps to the HSS $D_{SS}$ and, as we have seen above,
 $P=BK$ is the isotropy subgroup of the $1$-point cycle in $D_{SS}$, it follows in particular that $B$ stabilizes $\hat{C}_0$ and that it therefore  stabilizes $C_0$.
 In fact  $P:=B.K$ is its full stabilizer.  In this case $\mathcal{M}_Z$ is again $Z_{SS}=G/P$ and $\mathcal{M}_D$ is the HSS flag domain $D_{SS}$.

 \bigskip\noindent
 In Example 2 (general) the flag domain $D$ does not necessarily fiber over a HSS, but the phenomenon is more or less the same: $D$ lifts to the
 canonical domain $\hat{D}$ in the manifold of full flags which then maps down to the Hermitian symmetric space $D_{SS}$.  For the purposes of our work
 here it is only important that $\mathcal{M}_D$ is the Hermitian symmetric space  $D_{SS}$ and that $\mathcal{M}_Z=Z_{SS}$.   We then refer to such $D$ as those having
 $\mathcal{M}_D=D_{SS}$.

 \bigskip\noindent
 If $\mathcal{M}_D$ is not a HSS, e.g., if $G_0$ is not of Hermitian type, then, as indicated above, $G_{[C_0]}$ is at
 most a finite extension of $K$ (see Remark \ref{isotropy}).  In that setting we abuse notation and write $G_{[C]_0}=K$ with  $\mathcal{M}_Z=G/K$ and $\mathcal{M}_D$
 being the corresponding open subset.   Thus we have the following dichotomy.
 %
 \begin {proposition}
 Either $\mathcal{M}_D=D_{SS}$ as in the above examples or $\mathcal{M}_Z=G/K$.
 \end {proposition}

\noindent
Let us conclude this paragraph by formulating the above in terms of tangent spaces of cycle spaces.
Recall that if $\mathcal{M}_D=D_{SS}=G_0/K_0$, then
$T_{[C_0]}\mathcal{M}_Z$ is either isomorphic to $\mathfrak{s}^+$ or $\mathfrak{s}^-$ as a $K$-representation.  In particular, it is irreducible.
In the other case, i.e., where $\mathcal{M}_Z=G/K$, there are two possibilities:  If $G_0$ is not of Hermitian type, then the $K$-representation
on $T_{[C_0]}G/K$ is irreducible since $K$ is maximal subgroup of $G$.  However, if $G_0$ is of Hermitian type, then the cycle space $\mathcal{M}_D$ is the product of a $D_{SS}$ and its complex conjugate $\overline{D}_{SS}$ (\cite{BHH}).   In that case the representation of $K_0$ (resp. $K$) on the corresponding tangent space splits:
$T_{[C_0]}\cong \mathfrak{s}^-\oplus \mathfrak{s}^+$.

\section {Related results}
In (\cite{Hu}) the notion of cycle connectivity was considered:  A flag domain $D$ is said to be \emph{cycle connected} if given any two points
$p,q\in D$ there exists a connected chain $\mathcal{C}=C_0+\ldots +C_N$ with $C_i\in \mathcal{M}_D$ so that $p\in C_0$ and $q\in C_N$. In particular it
was shown that $D$ is not holomorphically convex if and only if it is cycle connected.
One says that $D$ is  $k$-cycle connected if independent of $p,q$ the number $N$ is at most $k$.  In the $1$-connected case relatively compact neighborhoods
$Y$ of the base cycle $C_0$ can be constructed in such a way that every point of the closure of $Y$ is contained in a cycle $C_1$ which is connected to $C_0$
(\cite{Hu}).  In this way one shows that $D$ is pseudoconcave of degree one.

\bigskip\noindent
In (\cite{K}), in an even more general cycle connected setting, a finiteness theorem at the level of $H^0$ was proved.  It is interesting that an important component
of the argument in (\cite{K}) is reminiscent of Andreotti's application of Siegel's Schwarz-Lemma method for proving finiteness theorems for pseudoconcave
complex spaces.

\bigskip\noindent
In further related work  (\cite{HHL}) we study flag domains from the point of view of cycle connectivity.  In particular, applications of the method of
connecting points of $D$ to the base cycle by closures of orbits of distinguished unipotent $1$-parameter groups are given.  Certain of these results,
which were motivated by applications to Hodge theory, appear in the unpublished preprint (\cite{Ha}) and others in the thesis (\cite{L}).  Finally, in
(\cite{HHS}) the Levi geometry of $D$ is studied from the point of view of curvature invariants of the normal bundle of $C_0$.  In particular $1$-pseudoconcavity
is proved by computing the Levi-form of boundary of a neighborhoods of $C_0$ which are defined as sublevel sets of functions which are transported norm
functions on the normal bundle of $C_0$.
\section {Supporting cycles}
 %
Here we construct the sets $Y$ which display the pseudoconcavity of a (non-holomorphically convex) flag domain $D$.
Every boundary point $p_0\in \partial Y$ is contained in at least one cycle $C$ which is contained in the closure $c\ell(Y)$ and which is
referred to as a supporting cycle at $p_0$.

\bigskip\noindent
In the setting of the previous section, let $\mathfrak{X}_Z:=\{(p,[C])\in Z\times \mathcal{M}_Z: p\in C\}$ and $\mu :\mathfrak{X}_Z\to Z$ and
$\nu :\mathfrak{X}_Z\to \mathcal{M}_Z$ be the projections to the individual factors.

The maps $\mu $ and $\nu $ define homogeneous holomorphic $G$-bundles and are in particular open maps. Since the neutral fiber of $\nu $ is just the base cycle $C_0$, it is
immediate that it is proper.  If we replace $Z$ by $D$ in this diagram, the resulting maps are still $G_0$-equivariant and open.  In this case it is known that $\mathcal{M}_D$
is a Stein manifold which is topologically a cell. Consequently, the bundle $\nu :\mathfrak{X}_D\to \mathcal{M}_D$ is trivial.

\subsection {Basic construction}
Here $B$ denotes any relatively compact domain with smooth boundary which contains the base point $[C_0]$ in $\mathcal{M}_D$.
Its preimage by the proper bundle map $\nu $ is denoted by $X$.  The relatively compact image in $D$ of $X$ by the open map $\mu $ is
denoted by $Y$.  Its closure $c\ell(Y)$ is the $\mu $-image of $c\ell(X)$.  If $p_0\in \partial Y$, then there exists $[C]$ in $\partial B$ with
$(p_0, [C])\in \partial X$, i.e., every boundary point of $Y$ is contained in a supporting cycle.
If $\mathcal{O}(D)\not\equiv \mathbb{C}$,
then $D$ fibers as a homogeneous bundle over a Hermitian symmetric space $\widehat{D}$ of non-compact type.  In fact the fibration $D\to \widehat{D}$
is another realization of $\nu:\mathfrak{X}_D\to \mathcal{M}_D$ and in that case $\mu $ is just the identity.  Thus in that case the supporting cycles
are completely contained in the boundary of $X$ and no pseudoconcavity observable in this construction.  On the other hand, we have the
following main result of this section.
\begin {theorem}
\label{supporting cycle intersection}
If $D$ is a flag domain with $\mathcal {O}(D)\cong \mathbb{C}$, i.e., which does not fiber holomorphically nor anti-holomorphically over a Hermitian symmetric space
of bounded domain type, then for every supporting cycle $C$ the intersection $C\cap Y$ is open and dense in $C$.
\end {theorem}
\noindent
The proof is a consequence of the results in the following paragraph.  Note that the intersection $C\cap \partial Y$ of a supporting cycle and $\partial Y$
could contain positive dimensional analytic sets. In fact in general this is the case.  However, one can always find a holomorphic disk $\Delta $
in $C$ which is at least $1$-dimensional and which displays the pseudoconcavity of at the given point.

\bigskip\noindent
\textit{Proof of Theorem \ref{weak pseudoconcavity}.}  Let $p_0\in \partial Y$ and $C$ be a supporting cycle at $p_0$.  Recall that $C$ is a flag manifold of
some conjugate of $K$.  Consequently there exists a Borel subgroup $B$ of $K$ with $B. p_0$ Zariski dense in $C$.  From the above theorem it then
follows that intersection $B.p_0\cap Y$ is non-empty.   If we identify $B.p_0$ with $\mathbb{C}^n$ we may choose a complex affine line $A$ which contains
both $p_0$ and $p \in B.p_0\cap Y$ when regarded as an affine curve in $B.p_0$.  The normalization $\hat{A}$ of the closure of $A$ in $Z$ is a copy of $\mathbb{P}_1(\mathbb{C})$.
If $U$ is a sufficiently small disk containing $p_0$, then $c\ell(U)\subset Y$ and $\Delta:=\hat{A}\setminus U$ is a holomorphic disk which displays the
concavity at the boundary point $p_0$.\qed

\subsection {Cycle space isotropy representations}
We now continue toward the proof of Theorem \ref{supporting cycle intersection}.
For this there are three cases that must be considered.  First, if $G_0$ is of Hermitian type, then $\mathcal{M}_D$ is either the associated symmetric space $D_{SS}$  or the
product $D_{SS}\times \overline{D}_{SS}$ of $D_{SS}$ with its complex conjugate.  If $G_0$ is not of Hermitian type, then,
just as in the case where $G_0$ is Hermitian and $\mathcal{M}_D=D_{SS}\times \overline{D}_{SS}$, the cycle space
$\mathcal{M}_D$ is a well-studied neighborhood of the real symmetric space $G_0.C_0\cong G_0/K_0$ in the affine symmetric space $\mathcal{M}_Z=G/K$.

\bigskip\noindent
The key ingredient for the main result in the following paragraph is that if $G_0$ is of Hermitian type and $\mathcal{M}_D\cong D_{SS}$ or $G_0$ is not of
Hermitian type, then the isotropy representation of $K$ on $T_{[C_0]}\mathcal{M}_D$ is irreducible.   Since the $[C]=g([C_0])$ for some $g\in G$, the same holds
for the $g$-conjugate $\widehat{K}$ which acts transitively on $C$.
If $G_0$ is of Hermitian type
and
$\mathcal{M}_D\cong D_{SS}\times \overline{D}_{SS}$,
then this representation is not irreducible. Handling this case requires closer attention.
\subsubsection {Supporting cycle property: The case of an irreducible isotropy representation}
Before going into the details of our construction, let us mention the following
basic fact.
\begin {lemma}
For every $p\in Z$ it follows that the isotropy group
$G_p$ acts transitively on $\{ [C]\in \mathcal{M}_Z: p\in C\}=\nu \mu^{-1}(p)$.
\end {lemma}
\begin {proof}
It is enough to prove this for $p\in C_0$.  For this let $C=g(C_0)$ with $p\in C\cap C_0$
and let $\hat{p}\in C_0$ with $g(\hat{p})=p$. Since $C_0$ is a $K$-orbit, there exists
$k\in K$ with $k(p)=\hat{p}$. If $\hat{g}:=gk$, then $\hat{g}(C_0)=C$ and $\hat{g}(p)=p$.
\end {proof}

Now, turning to the construction of supporting cycles,  if $p_0\in \partial Y$, then there exists $[C]\in \partial B$ with $(p_0,[C])\in \partial X$.
For $p\in C$ we consider the set $\nu\mu^{-1}(p)$ which by the above Lemma can be described as $\{[\hat{C}]:p\in [\hat{C}]\}=G_{p_0}([C])$.
In particular it is a smooth complex submanifold of $\mathcal{M}_D$ at $[C]$.
We wish to compare the tangent space $T_{[C]}G_p.[C]$ and the complex tangent space $H_{[C]}$ of $\partial B$ at $[C]$.   For this recall that
$H_{[C]}$ is the maximal complex vector subspace of the real tangent space to the boundary $\partial B$ and is a $1$-codimensional complex
subspace of the full tangent space $T_{[C]}\mathcal{M}_D$.

\bigskip\noindent
The following is a restatement of Theorem \ref{supporting cycle intersection} in the context of the present paragraph.
\begin {proposition} \label{irreducible}
Let $D$ be a flag domain which possesses no non-constant holomorphic functions and let $C$ be a boundary supporting cycle at $p_0\in \partial Y$.
Then, if $G_0$ is not of Hermitian type or is of Hermitian type with $\mathcal{M}_D\cong D_{SS}$, it follows that $C\cap Y$ contains an open dense
subset of $C$.
\end {proposition}
\begin {proof}
If for $p\in C$ the stabilizer of $C$ contains the isotropy group $G_p$, then it contains $G_pK$ and is therefore a parabolic subgroup $P$ of $G$.
Consequently, the map $G/G_p\to G/P$ maps $D$ to a flag domain where the base cycle is just a point, i.e., to a HSS.  Since this is not the case
under the assumptions of the proposition, it follows that $G_p[C]$ is non-trivial with tangent space $V_p\not=0$
contained in the irreducible $\widehat{K}$-representation space $T_{[C]}\mathcal{M}_D$.  Since the
complex tangent space $H_{[C]}$ is \emph{not} $\widehat{K}$-invariant, it follows that that there is a dense open subset
of $C$ of points of the form $p=\widehat{k}(p_0)$ with $V_p\not\subset H_{[C]}$, i.e., $V_p+H_{[C]}=T_{[C]}\mathcal{M}_D$.
For such $p$ it follows that $V_p$ is transversal to $T_{[C]}\partial{B}$ and therefore $G_p.[C]\cap B\not=\emptyset $.  Equivalently, the
$\mu $-fiber at such a point has non-empty intersection with $X$.  In other words, $p\in Y$.
\end {proof}

\subsubsection {Supporting cycle property: The Hermitian case where $\mathcal{M}_D\cong D_{SS}\times \overline{D}_{SS}$}
Our goal here is to prove the analogous result to Proposition 2.3 in the case where $G_0$ is of Hermitian type and
$\mathcal{M}_D=D_{SS}\times \overline{D}_{SS}$.   Here $\mathcal{M}_Z\cong G/K$, but the isotropy representation of $K$
is not irreducible.   Focusing on the supporting cycle $[C]=g([C_0])$ and the $g$-conjugate $\widehat{K}$ which stabilizes it, we
choose a maximal torus $\widehat{T}$ of $G$ which is contained in $G_{p_0}\cap \widehat{K}$.  Carrying out a root decomposition
with respect to $\widehat{\mathfrak{t}}$, it follows that
$\mathfrak{g}=\widehat{\mathfrak{u}}_-\oplus \widehat{\mathfrak{k}}\oplus \widehat{\mathfrak{u}}_+$ where
$\widehat{\mathfrak{p}}_\pm =\widehat{\mathfrak{k}}\oplus \widehat{\mathfrak{u}}_\pm $ are the standard parabolic algebras containing
$\widehat{\mathfrak{k}}$.  Since $\widehat{T}\subset G_{p_0}$ the tangent space $T_{[C]}G_{p_0}.[C]$ is then realized as a subspace $V_{p_0}$
 which is a direct sum of certain $\widehat{t}$ root algebras in
$\widehat{\mathfrak{u}}_-\oplus \widehat{\mathfrak{u}}_+$ .  If $V_{p_0}$ were
contained in one or the other of $\widehat{\mathfrak{u}}_\pm $, then $G_{p_0}$ would be contained in one or the other of the standard
parabolic subgroups $P_\pm $.  Denoting this parabolic group by $P$, this would result in a fibration $G/G_{p_0}\to G/P$.  But the
$G_0$-orbit at the point $P$ in the base is an associated Hermitian symmetric space of bounded type.  This is the essential observation
for the proof of the following version of Theorem \ref{supporting cycle intersection} in the context of the present paragraph.
\begin {proposition}
If $D$ is a $G_0$-flag domain with $\mathcal{O}(D)\cong \mathbb{C}$ and $G_0$ is of Hermitian type with
$\mathcal{M}_D\cong D_{SS}\times \overline{D}_{SS}$,  then every supporting cycle $C$ has the property
that $C\cap Y$ is an open dense subset of $C$.
\end {proposition}
\begin {proof}
The $\widehat{K}$-representations on $\widehat{\mathfrak{u}}_+$ and $\widehat{\mathfrak{u}}_-$ are irreducible.
As was remarked above, since $\mathcal{O}(D)\cong \mathbb{C}$, it follows that both of these nilradicals have non-trivial
intersection with $V_{p_0}$.   Thus $Span\{\widehat{k}(V_{p_0}): \widehat{k}\in \widehat{K}\}$ is the full tangent space
$T_{[C]}\mathcal{M}_D\cong \widehat{\mathfrak{u}}_-\oplus \widehat{\mathfrak{u}}_+$.  Since $\widehat{k}(V_{p_0})=V_{\widehat{k}(p_0)}$,
it follows that for any given complex hyperplane $H$ in $T_{[C]}\mathcal{M}_D$ the set of points $p=\widehat{k}(p_0)$ in $C$
for which $H+V_{p}=T_{[C]}\mathcal{M}_D$ is open and dense in $[C]$.  As in the previous case, this transversality
implies that $p\in Y$.
\end {proof}
\section {A lower bound on the pseudoconcavity}
Let $Y$ be as above and denote by $\partial Y$ the boundary of its interior. If $p_0\in \partial Y$, then
every supporting cycle $C$ at $p_0$ displays a certain degree $c(C,p_0)$ of pseudoconcavity of $Y$ at $p_0$
in the sense that there is a $c(C,p_0)$-dimensional polydisk $\Delta $ contained in $C$ and centered at $p_0$
with the property that $\Delta \setminus \{p_0\}$ is contained in $C$.  Furthermore, $c(C,p_0)$ is maximal
with this property.  Then the \emph{supporting cycle pseudoconcavity} of $Y$ at $p_0$ is defined to
be the maximum the $c(C,p_0)$ as $C$ runs through all supporting cycles containing $p_0$.  Finally,
the pseudoconcavity of $Y$ is defined to be the minimum $c(Y)$ of
the $c(p_0)$ as $p_0$ runs through $\partial Y$.  As the reader will note, this pseudoconcavity has
very little to do with the exact nature of $Y$, i.e., with the choice of the domain $B$ in $\mathcal{M}_Z$.  In any case the estimate
$c(Y)>d_{ma}$(Theorem \ref{main} and Corollary \ref{pseudoconcave}), which we prove below in the equal rank case, is in terms of the root theoretic invariant
$d_{ma}$ which only depends on the closed $K$-orbit in $Z$. \\

Let us emphasize that we restrict here to the equal rank case, i.e., where the maximal tori of $K$ are also maximal
tori of $G$.  Let $p_0\in \partial Y$ and $[C]$ be a supporting cycle in $c\ell(Y)$ at $p_0$.  The
$G$-isotropy group at $[C]$ is either the appropriate conjugate of $K$ or of a certain
parabolic subgroup $P$ of $G$ which contains $K$.  We denote this group by $J$ and
let $H$ be a complex hyperplane in $\mathfrak{g}/\mathfrak{j}$.  In our discussion here
the conjugation plays no role at all and consequently we may assume that $\mathfrak{j}$
is either $\mathfrak{k}$ or $\mathfrak{p}$, depending on the case at hand.   Both contain
a maximal  toral subalgebra $\mathfrak{t}$ which, due to our assumption of maximal rank, is a
maximal torus in both $\mathfrak{g}$ as well.  The root decompositions which are used
below are with respect to $\mathfrak{t}$. For $\delta $ a $\mathfrak{t}$-root we let $e_\delta $
be such that $\mathbb{C}e_\delta =\mathfrak{g}_\delta $.

\bigskip\noindent
The hyperplane $H$ will be viewed as being in $\mathfrak{g}$ containing $\mathfrak{j}+\mathfrak{q}_0$
where $\mathfrak{q}_0=Lie(Q_0)$ and $Q_0$ is the $G$-isotropy group $G_{p_0}$.  We let
$\oplus \mathfrak{g}_\alpha=:A$, $\alpha \in \Phi $,  be the $\mathfrak{t}$-invariant complement
of $\mathfrak{j}+\mathfrak{q}_0$ with $(e_\alpha :\alpha \in \Phi )$ its basis. Denote its dual basis by
$(f_\alpha:\alpha\in \Phi)$ and let $L=\sum a_\alpha f_\alpha $ be a linear function which has $H$
as its $0$-set.

\bigskip\noindent
Given $p\in C$ we are interested in whether or not the orbit $G_p.[C]$ is tangent to $H$.  For our purposes
it is only necessary to discuss this matter locally near $p_0$; in particular with $p\in U^-.p_0$ where $U^-$ is the opposite
of the unipotent radical of the isotropy group $K_{p_0}$.  For this we identify this orbit with $\mathfrak{u}^-=\oplus \mathfrak{g}_\beta $
for $\beta $ running in the roots $\Lambda (\mathfrak{u}^-)$.  Finally, we let $E:=\oplus \mathfrak{g}_\gamma$, $\gamma \in \Gamma $, be
the $\mathfrak{t}$-root complement of $\mathfrak{j}\cap \mathfrak{q}_0$ in $\mathfrak{q}_0$.
If $exp(\xi )\in U^-$ with $p=Ad(exp(\xi)).p_0$, then $T_{[C]}G_p.[C]\subset H$ if and only if $L(Ad(exp(\xi))(E)))=0$.  Let us refer to
$\{\xi \in A: Ad(exp(\xi))(E)))=0\}$ as the \emph{boundary variety} at $p_0$ and denote it by $S_L$.
Since $Ad(exp(\xi))=e^{ad(\xi)}$, the system of defining equations for $S_L$ which consists of somewhat complicated non-linear equations and is
not easily understandable.Therefore we will be satisfied with determining a linear outer approximation $\widehat{S}_L$ which contains $S_L$.

\subsection {Linearization}
Let $X$ be a vector field defined by a $1$-parameter group $g_X(t)=exp(\xi t)$ with $\xi=\sum \xi_\beta e_\beta \in \mathfrak{u}^-$.
The $g_X(t)$-conjugate of the $G_{p_0}$-orbit at $[C]$ is no longer in $H$ if $[X,W]\not\in H$ for some field $W$ corresponding
to an element of $\eta \in E$.  In other words, $Ad(exp(\xi t))(\eta )$ is a curve which satisfies $L(Ad(exp(\xi t)(\eta )))\not=0$ for $t\not=0$
sufficiently small.   Thus if $g_X(t)$ maps the tangent space to the $G_{p_0}$-orbit to a space which is still in $H$, then
$L(exp(\xi t)(\eta ))=0$ for all $\eta \in E$.   Since $L$ is linear, we have an elementary linear approximation.
\begin {proposition}
In the coordinates defined by $\mathfrak{u}^-$ the boundary variety $S_L$ is contained in
$$
\widehat{S}_L:=\{\xi \in A: L([\xi,\eta])=0 \ \text{for all}\ \eta \in E\}\,.
$$
\end {proposition}
\begin {proof}
$$
\frac{d}{dt}\Big\vert_{t=0}L(Ad(exp(t\xi)(\eta)))=L(\frac{d}{dt}\Big\vert_{t=0}Ad(exp(t\xi)(\eta)))=L([\xi,\eta])\,.
$$
\end {proof}
\subsection {On the maximum dimension of $\widehat{S}_L$}
For a generic linear function $L$ it is difficult to compute $\widehat{S}_L$.  However, for $L=L_\alpha =f_\alpha $ this job can
be translated to a manageable root-combinatoric.  The following fact is therefore quite useful.
\begin {proposition}\label{maximum}
There exits $\alpha \in \Phi$ such that
$$
max_{L}\{dim_\mathbb{C}\widehat{S}_L\}=dim_\mathbb{C}S_{L_\alpha}\,.
$$
\end {proposition}

\noindent
Preparing for the proof,  we first note that in this setting we can view $L$ as varying in projective space so that the maximum is taken on.
Secondly, since $E$ is $T$-invariant, for $t\in T$ it follows that
$$
t(L)([t(\xi),\eta])=0\ \text{for all} \ \eta \in E \ \Longleftrightarrow \ L([\xi,\eta ])=0 \ \text{for all}\ \eta \in E\,.
$$
Thus $\widehat{S}_{t(L)}=t(\widehat {S}_L)$.\\

\noindent
\textit{Proof of Proposition \ref{maximum}}\\
Let $\eta_1,\ldots \eta_n$ be any basis of $E$ and for $L$ given define $\varphi_L:A\to \mathbb{C}^n$ by
$\xi\mapsto (L([\xi,\eta_1]),\ldots ,L([\xi, \eta_n]))$.  Then let $\mathcal{L}$ be the space of all $L$ and define
$\mathcal{X}:=\mathcal{L}\times A$ with $\varphi :\mathcal{X}\to \mathbb{C}^n$ defined by
$\varphi (L,\xi)=\varphi_L(\xi)$.  Finally,  let $\mathcal{K}:=\varphi^{-1}(0)$
and consider the $T$-equivariant surjective projection $\mathcal{K}\to \mathcal{L}$.  Assume that the maximum
dimension $d$ is taken on for the function $L$.  It follows from Proposition \ref{maximum} that $dim_\mathcal{C}S_{t(L)}=d$ for all $t\in T$.
Now let $t(L)$ tend to a point $L_0$ which (in the projective space) is $T$-fixed.  By semicontinuity of fiber
dimension it follows that $dim_\mathcal{C}S_{L_0}\ge d$.  But by the choice of $d$ being maximal, equality
follows.  Consequently, the maximum dimension is taken on by a function $L_0$ with is projectively a $T$-fixed
point  But $L_0=\sum a_\alpha L_\alpha $ transforms by $t(L_0)=\sum a_\alpha \chi_\alpha(t)L_\alpha $ where
$\chi_\alpha $ is the character associated to the root $\alpha $.  Since $\chi_\alpha=\chi_\beta $ if and only if
$\alpha=\beta $, it follows that $L_0=L_\alpha $ for some $\alpha $.\qed
\subsection {Lower bound of pseudoconcavity}
The \emph{attractiveness} of  root $\alpha \in \Phi $ is defined to by
$$
At(\alpha)=\{\beta\in \Lambda (\mathfrak{u}^-): \ \text{there exists}\ \gamma \in \Gamma \ \text{with}\ \gamma +\beta =\alpha\}\,.
$$
Note that if $At(\alpha)=\{\beta_1,\ldots ,\beta_N\}$, then there are unique pairwise different
$\gamma_1,\ldots ,\gamma_N$ with $\alpha=\beta_i+\gamma_i$, $i=1,\ldots ,N$.  Let $R_\alpha$ be the subspace
of $A$ (regarded in the cycle $C$) which is the span of the $e_{\beta_i}$.  By construction it follows that
$R_\alpha $ has trivial intersection with $\widehat{S}_{L_\alpha}$.  Since the $\beta_i$ are independent, $dim_\mathbb{C}R_\alpha =\vert At(\alpha)\vert$.
Going through the logic, we then have that the maximal dimension $max_L\{dim_\mathbb{C}{S}_L\}$ is achieved by some $S_{L_\alpha}$.
Its dimension is at most that of $\widehat{S}_{L_{\alpha}}$ which is in turn less than or equal to the
$codim_\mathbb{C}R_\alpha $.  Thus we let $\alpha_{min}$ be any root in $\Phi $ with minimal attractiveness and define
$d_{ma}:=\vert At(\alpha_{min})\vert$.
\begin {theorem} \label{main}
For every $L\in \mathcal{L}$ it follows that $codim_\mathbb{C}S_L\ge d_{ma}\,.$
\end {theorem}
\begin {proof}
Since
$
R_\alpha \cap S_{L_{\alpha}}=0
$
and $d_{ma}\le dim_\mathbb{C}R_\alpha $ for all $\alpha $, the proof follows immediately from the above results.
\end {proof}

\noindent
Note that $d_{ma}$ is an invariant which only depends on the $K$-orbit $C_0$ in $Z$, i.e., not on the choice of the domain
$B$ in $\mathcal{M}_Z$ at the outset of the construction.
\begin {corollary}\label{pseudoconcave}
The pseudoconcavity $c(Y) $ is at least the dimension of the span $R_\alpha $ of the root spaces
$\mathfrak{g}_\beta $ for $\beta \in At(\alpha )$ where $\alpha $ is a root of minimum attractiveness, i.e.,
$c(Y)\ge d_{ma}\,.$
\end {corollary}
\noindent
\textbf{Remark.} In closing we should remind the reader that all results in this paragraph were proved under the assumption
that $rank(G)=rank(K)$.

\section{Explicit lower bounds in Grassmannians}
Here we compute explicit lower bounds for the pseudoconcavity of flag domains in Grassmannians in
the equal rank cases of the mixed signature unitary and real symplectic groups.   Although it is possible
that the stabilizer of the base cycle is a parabolic group $P$ containing $K$, since the methods for that
case are exactly the same as those which the stabilizer is $K$, we leave that case to the reader.
\subsection{$SU(p,p^\prime)$-orbits in Grassmannians}
Let $SU(p,p^\prime)$ denote the real form of $SL_{n}(\mathbb{C)}$ consisting
of transformations which preserve the mixed signature Hermitian form given by
$h(z,w)=z^tE_{p,p^\prime}\bar{w}$ where $E_{p,p^\prime}=Diag(Id_p,-Id_{p^\prime})$.
The flag domains in the Grassmannian
$Z:=Gr_k(\mathbb{C}^n)$ of $k$-dimensional subspaces of $\mathbb{C}^n$ are
parameterized by pairs $(r,r^\prime)$ of non-negative integers with $r\le p$ and $r^\prime \le p^\prime $:
$D_{r,r^\prime}=\{V\in Z: sign(h\vert V)=(r,r^\prime)\}$.

\bigskip\noindent
As we have seen above, we may choose $K$ in any convenient form and $C_0$ to be its base cycle.
Thus we choose $K$ to have matrices in block form which looks like
\begin {gather*}
\begin {pmatrix}
K_1 & 0\\
0 & K_2
\end {pmatrix}
\end {gather*}
where $K_1$ (resp. $K_2$) is a $p\times p$-matrix (resp. $p^\prime\times p^\prime$-matrix) in a standard basis $(e_1,\ldots ,e_p,f_1,\ldots ,f_{p^\prime})$.
In the above estimate, the dimension of $S_H$ is maximized over all $H$ and consequently we may choose base point $p_0\in C_0$ at our convenience.
We do so by letting $p_0:=Span\{e_1,\ldots, e_r,f_1,\ldots ,f_{r^\prime}\}$.  The unipotent group that we will use
to move $p_0$ to nearby points looks like
\begin {gather*}
K_1=
\begin {pmatrix}
Id_{r\times r} & 0\\
A  & Id_{(p-r)\times (p-r)}
\end {pmatrix}
\ \text{and} \ K_2=
\begin {pmatrix}
Id_{s\times s} & 0\\
B & Id_{(p^\prime-r^\prime)\times (p^\prime-r^\prime)}
\end {pmatrix}
\end {gather*}
where $A$ and $B$ are arbitrary matrices of the appropriate sizes.  Of course the $K$-isotropy group at the base point is
complementary to these groups.  In addition to the part in $K$ the isotropy group $Q_0$ contains the unipotent group of matrices of the form
\begin {gather*}
\begin {pmatrix}
Id_{p\times p} & U_+\\
U_- & Id_{p^\prime \times p^\prime}
\end {pmatrix}
\end {gather*}
where for example  $U_+$ consists of matrices of the form
\begin {gather*}
\begin {pmatrix}
a & b\\
0 & c
\end {pmatrix}
.
\end {gather*}
Here the entries of the matrices $a $, $b$ and $c$ are arbitrary complex numbers.

\bigskip\noindent
Now our job is to conjugate $U_+$ and $U_-$ with matrices complementary to the isotropy group $K_{p_0}$
and determine what is obtained in the block, e.g., of the $U_+$-picture, where above there is $0$.  For example
for $U_+$ we have
\begin{gather*}
K_1U_+K_2^{-1}=
\begin {pmatrix}
Id & 0\\
A & Id
\end{pmatrix}
\begin{pmatrix}
a & b\\
0  & c
\end{pmatrix}
\begin{pmatrix}
Id & 0\\
-B & Id
\end{pmatrix}
\end{gather*}
so that the relevant block is $A(a -bB)-cB=R$ where $a$,$b$ and $c$ are arbitrary.   For $U_-$ we have an analogous block.

\bigskip\noindent
Now let $\mathfrak{g}_{ij}$ be a spot in the block of relevance of either $U_+$ or $U_-$, i.e., where originally we had $0$.
The above general result shows that the minimum of the $At(\alpha)$ will be achieved when the linear defining function $f_\alpha $
is just the evaluation at such a spot.  If that $\alpha $ is this root, then the set $S(C)_\alpha $ is defined by the condition
$R_{ij}=0$ for all values of $a$, $b$ and $c$.  For example, this implies that $(Aa)_{ij}=0$ for all $a$ or
equivalently $A_{ik}a_{kj}=0$ for all $a_{kj}$.  In other words the $i$-th row of $A$ vanishes.   Analogously it follows that the $j$-th column of $B$
vanishes.

\bigskip\noindent
The above discussion was for $\mathfrak{g}_{ij}$ and an upper-triangular spot.   For a lower triangular spot the roles of $A$ and $B$ are reversed.
Thus we have the following remark.
\begin {proposition}
If $f_\alpha $ is the root  functional defined by the spot $\mathfrak{g}_{ij}$, then
$codim (S(C)_\alpha)$ is either the length of $A_{i*}$ plus the length of $B_{*j}$ or vice versa.
\end {proposition}
\noindent
In the case where $\mathfrak{g}_{ij}$ is in the upper-triangular piece,  the total codimension (vanishing spots in $A$ and $B$) is then
$r+p^\prime-r^\prime$.  In the opposite case it is $p-r+p^\prime$.  Thus, applying the estimate derived in the previous paragraph, we
have the following result.
\begin {corollary}
For the $SU(p,p^\prime)$-flag domain $D_{r,r^\prime}$ it follows that
$$
c(Y)\ge min\{r+p^\prime-r^\prime, p-r+p^\prime\}\,.
$$
\end {corollary}

\noindent
\textbf{Remarks.} The above was carried out under the assumption that $\Phi$ parameterizes the complement of $\mathfrak{k}\oplus \mathfrak{q}_0$.
In the case where the stabilizer algebra $\mathfrak{h}$ is a larger parabolic algebra, one obtains a much sharper estimate.  Note that in the cases where
$D$ is a HSS, i.e., $D_{r,0}$ and $D_{0,r^\prime}$, lower concavity bound is, as it should be, zero. This is due to the fact that  $\mathfrak{u}^-=0$.

\subsection{$Sp(2n, \mathbb{R})$-orbits in Grassmannians}
Let $G_0=Aut{(\mathbb{R}^{2n},h)}\cong Sp(2n,\mathbb{R})$ with a non-degenerate alternating form $h$.
Let $Z$ be a Grassmannian of isotropic $k$-planes in $\mathbb{C}^{2n}$  with $k<n$ and let $D$ be the open $G_0$-orbit of planes of signature $(p,q)$ with $p+q=k$.
We may choose a basis $v_1,\cdots ,v_{2n}$ of $\mathcal{C}^{2n}$ satisfying $v_j=\overline{v_{n+j}}$ and $h(v_j,v_{n+k})=\delta_{jk}$
for $j\leq n$ so that
$$p_{0}=Span_{\mathbb{C}}\{v_1,\cdots, v_p,v_{2n-q+1},\cdots,v_{2n}\}$$
is a base point in $D$.
Let $\mathfrak{q}_0$ be the isotropy subalgebra of $\mathfrak{g}$ at $p_0$.
Then we have
\begin{gather*}
\mathfrak{k}=
\begin{pmatrix}
k&0\\
0&-{}^tk\\
\end{pmatrix}
\ \text{for}\ k\in \mathop{\mathrm{Mat}}\nolimits(n,n).
\end{gather*}
The nilpotent subalgebra $\mathfrak{u}^-$ is the subalgebra of matrices of the form
\begin {gather*}
k=
\begin {pmatrix}
0 & 0 & 0\\
C_1 & 0 & 0\\
C_2 & C_3 & 0
\end {pmatrix}
\end {gather*}
where
$C_1\in\mathop{\mathrm{Mat}}\nolimits(n-p-q,p)$ , $C_2\in\mathop{\mathrm{Mat}}\nolimits{(q,p)}$ and $C_3\in\mathop{\mathrm{Mat}}\nolimits{(q,n-p-q)}$.

\bigskip\noindent
We then have the decomposition $\mathfrak{g}=\mathfrak{s}^+\oplus \mathfrak{k}\oplus \mathfrak{s}^-$ where
\begin{align*}
&\mathfrak{s}^+=\left\{\begin{pmatrix}
0&A^+\\
0&0\\
\end{pmatrix};\; A^+\in \mathop{\mathrm{Sym}}\nolimits{(n)}\right\},\\
&\mathfrak{s}^-=\left\{
\begin{pmatrix}
0&0\\
A^-&0\\
\end{pmatrix};\; A^-\in \mathop{\mathrm{Sym}}\nolimits{(n)}\right\}
\end{align*}
and the parabolic subalgebra $\mathfrak{q}_0$ decomposes into
$$\mathfrak{q}_0=(\mathfrak{s}^+\cap\mathfrak{q}_0)\oplus (\mathfrak{k}\cap\mathfrak{q}_0)\oplus (\mathfrak{s}^-\cap\mathfrak{q}_0).$$
Here $\mathfrak{q}_0\cap \mathfrak{s}^+$ is the subalgebra of matrices of the form given by
$$
A^+=
\left(
\begin{array}{cc:c}
&&A^+_2\\
\multicolumn{2}{c}{\mbox{\smash{\LARGE $A^+_1$}}}&0\\ \hdashline
{}^tA^+_2&0&0
\end{array}
\right)
\quad \text{where }
\begin{array}{l}
A^+_1\in\mathop{\mathrm{Sym}}\nolimits(n-q),\\
A^+_2\in\mathop{\mathrm{Mat}}\nolimits{(p,q)}.
\end{array}$$
Thus $\mathfrak{q}_0\cap \mathfrak{s}^-$ is the subalgebra of matrices of the form given by
$$
A^-=
\left(
\begin{array}{c:cc}
0&0&A^-_2\\
\hdashline
0&&\\
{}^tA^-_2&\multicolumn{2}{c}{\mbox{\smash{\LARGE $A^-_1$}}}
\end{array}
\right)
\quad \text{where }
\begin{array}{l}
A^-_1\in\mathop{\mathrm{Sym}}\nolimits(n-p),\\
A^-_2\in\mathop{\mathrm{Mat}}\nolimits{(p,q)}.
\end{array}$$
Let $v^k_{\ell}=v^k\otimes v_{\ell}\in (\mathbb{C}^{2n})^*\otimes \mathbb{C}^{2n}$ and let $e_k$ be the dual of $v^k_k-v^{n+k}_{n+k}$ for $1\leq k\leq n$.
Then the set of the roots with respect to the maximal diagonal subalgebra is
$$\Lambda(\mathfrak{g})=\{\pm e_j\pm e_k, \pm2e_{\ell}\;|\; 1\leq j< k\leq n,\; 1\leq \ell\leq n \}.$$
Denoting  $\pm (e_j+ e_k)$ (resp. $e_j-e_k$) by $e^\pm_{j,k}$ (resp. $e^c_{j,k}$) and we have
\begin{align*}
&\Lambda(\mathfrak{s}^{\pm})=\{e^{\pm}_{j,k}\}_{j\leq k}\\
&\Lambda(\mathfrak{s}^+\cap \mathfrak{q}_0)=
\left\{\begin{array}{l|l}
e^+_{j,k}&
\begin{array}{l}
1\leq j\leq k\leq n-q
\end{array}
\end{array}\right\}
\cup\left\{\begin{array}{l|l}
e^+_{j,k}&
\begin{array}{l}
1\leq j\leq p,\\
n-q< k\leq n
\end{array}
\end{array}\right\}\\
&\Lambda(\mathfrak{s}^-\cap \mathfrak{q}_0)=
\left\{\begin{array}{l|l}
e^-_{j,k}&
\begin{array}{l}
p< j\leq k\leq n
\end{array}
\end{array}\right\}
\cup\left\{\begin{array}{l|l}
e^-_{j,k}&
\begin{array}{l}
1\leq j\leq p,\\
n-q< k\leq n
\end{array}
\end{array}\right\}\\
&\Lambda(\mathfrak{u}^-)=
\left\{\begin{array}{l|l}
e^c_{j,k}&
\begin{array}{l}
p< j\leq n-q,\\
1\leq k\leq p
\end{array}
\end{array}\right\}
\cup\left\{\begin{array}{l|l}
e^c_{j,k}&
\begin{array}{l}
n-q< j\leq n,\\
1\leq k\leq n-q
\end{array}
\end{array}\right\}
\end{align*}
We set $\Gamma$, $\Phi$, and $At(\alpha)$ for $\alpha\in \Phi$ as the previous section.
\begin{lemma}\label{card}
$\vert At(\alpha)\vert \geq\begin{cases}n-q&\text{if }\alpha\in\Phi\cap\Lambda(\mathfrak{s}^+),\\ n-p&\text{if }\alpha\in\Phi\cap\Lambda(\mathfrak{s}^-)\end{cases}$.
\end{lemma}
\begin{proof}
If $\alpha=e_{\ell ,m}^+\in\Phi\cap\Lambda(\mathfrak{s}^+)$, then $n-q< m\leq n$ and $p<\ell\leq m$.
In this case, $\beta=e_{j,k}^c\in\Lambda(\mathfrak{u}^-)$ is contained in $At(\alpha)$ if $j= \ell$ or $m$.
A long root $\alpha=e_{\ell,\ell}^+$ minimize $\vert At(\alpha)\vert$ where
$$
At(\alpha)=\{e_{\ell ,1}^c,\cdots ,e_{\ell ,n-q}^c\}.
$$
If $\alpha\in\Phi\cap\Lambda(\mathfrak{s}^-)$, we analogously have
$$
At(\alpha)=\{e^c_{p+1,\ell},\cdots ,e^c_{n,\ell}\}
$$
for a long root $\alpha=e_{\ell,\ell}^-$ which minimize $\vert At(\alpha)\vert$.
\end{proof}
\begin{corollary}
$
c(Y)\ge \min{\{n-p,n-q\}}
$
\end{corollary}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction.}
\subsection{Main Result.}
For $\alpha\in (0,1)\backslash \mathbb{Q}$, denote the continued fraction expansion of $\alpha$ by
\begin{equation*}
\alpha = \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{\dots}}} = [a_1, a_2, \dots , a_n, \dots ]
\end{equation*}
where $a_n \in \mathbb{N}_+$ are the entries of the continued fraction and $\{ p_n/q_n \}_{n\in \mathbb{N}_+}$ are the convergents of $\alpha$, i.e. $ p_n/q_n  =  [a_1, a_2, \dots , a_n ]$ with $(p_n,q_n)=1$ .

We prove the following theorem.
\begin{teo}\label{main}
Given $R>0$, introduce
\begin{equation*}
n_R = \min \{ n \in \mathbb{N} \, | \, \,  q_n > R \} .
\end{equation*}
Fix also $N\geq0$. Then the ratio $\frac{q_{n_R}}{R}$ and the entries $a_{n_R-k}$ for $0\leq k < N$
 have a joint limiting distribution, as $R$ tends to infinity, with respect to the Gauss measure $\mu_{1}$ given by the density $\frac{\mathrm{d} \mu_1 }{\mathrm{d} \alpha}=(\ln 2 (1+\alpha))^{-1}$.
\end{teo}
Theorem \ref{main} means that for each $N\geq0$ there exists a probability measure $P_N$ on $(1,\infty) \times \mathbb{N}_+^N$ such that for all $a, b >1$, $c_k\in \mathbb{N}_+$, $0\leq k < N$,
\begin{equation} \label{limiting}
\begin{split}
 \mu_1  \{ \alpha  \, : \,  a < \frac{q_{n_R}}{R} < b,  \quad a_{n_R-k} = c_k, \, 0\leq k & < N \}  \xrightarrow{R\rightarrow \infty}
\\& P_N \left(
(a,b) \times \{ c_0 \} \times \dots \times \{ c_{N-1} \} \right).
\end{split}
\end{equation}
In  Theorem \ref{main}, instead of $\mu_1$, one can consider any absolutely continuous measure, but we do not dwell on this.

\subsection{Applications.}
Theorem \ref{main} is useful in many applications. As an example, we refer to the two following papers.
In \cite{BS:lim}, the authors consider the problem of limiting behavior of larges  Frobenius numbers, initially investigated by V.I.~Arnold \cite{Ar:wea}. If $a=(a_1,\dots, a_n)$ is an $n$-tuple of positive integers which are coprime, the Frobenius number $F(a)$ is the smallest $F$ such that any integer $t\geq F$ can be written in the form $t=\sum_{j=1}^n x_j a_j$ where $x_j$ are non-negative integers. Let $\Omega_N$ be the ensemble of all coprime $n$-tuples with entries less than $N$ with the uniform probability distribution.
In the case $n=3$, the existence of the limiting distribution of $\frac{1}{N^{3/2}}F(a)$ as $N$ tends to infinity is proved  using a discrete version of Theorem \ref{main}.

In \cite{SU:lim}, the following trigonometric sums are considered:
\begin{equation*}
\frac{1}{N}\sum_{n=0}^N  \frac{1}{1- e^{2\pi i (n\alpha + x)} },\qquad (x,\alpha)\in (0,1)\times(0,1),
\end{equation*}
where $(0,1)\times(0,1)$ is endowed with the uniform probability distribution. The authors prove that such trigonometric sums (and, more generally, the Birkhoff sums of a function with a singularity of type $1/x$ over a rotation) have a non-trivial joint limiting distribution in $x$ and $\alpha$ as $N$ tends to infinity.
Also in this case the  proof of the existence of the limiting distribution is based on the existence of the limiting distribution in Theorem \ref{main}.

\subsection{Outline.}
The main idea of the proof of Theorem \ref{main} is to reformulate the problem in terms of a certain special flow over the natural extension of the Gauss map and to exploit mixing of the flow to prove the existence of the limiting distribution. The definitions of the natural extension of the Gauss map and of special flow are recalled in \S \ref{definitionsection}. The reduction to a special flow is shown in \S\ref{reductionsec} and the existence of the limiting distribution is proved in \S\ref{limitingsec}.  The same special flow was considered also in \cite{DS:sta} and the proof that the special flow is mixing is recalled in \S \ref{mixingsection}.
\section{Definitions.}\label{definitionsection}
\subsection{Gauss map.}
Let $\mathscr{G}$ be the Gauss map, i.e. the transformation on $(0,1)$ given by $\alpha\mapsto \mathscr{G}(\alpha)= \{ \frac{1}{\alpha} \}$, where $\{\cdot \} $ denotes the fractional part. The Gauss measure $\mu_1$
 is invariant under $\mathscr{G}$.
 The sequence $\{a_n\}_{n\in \mathbb{N}_+}$ can be seen as a symboling coding for the orbit $\{ \mathscr{G} ^n \alpha \}_{n\in \mathbb{N}}$, since $a_n = [(\mathscr{G} ^{n-1} (\alpha))^{-1}]$ where $[\cdot]$ denotes the integer part.
A point $\alpha \in (0,1)\backslash \mathbb{Q}$ will be often identified with the infinite sequence $\{a_n\}_{n\in \mathbb{N}_+}$ in $\mathbb{N_+}^{\mathbb{N}_+}$.

For convenience, we will also use the following notation:
\begin{equation*}
[a_0; a_1, \dots, a_n , \dots ] = a_0 + \frac{1}{a_1 + \frac{1}{ \ddots}}, \qquad [a_0; a_1, \dots, a_n ] = a_0 + \frac{1}{a_1 + \frac{1}{\ddots + \frac{1}{a_n}}}.
\end{equation*}

\subsection{Natural extension of the Gauss map.}
The natural extension $\h{\mathscr{G}}$ of the Gauss map $\mathscr{G}$ acts on infinite bi-sided sequences $\{a_n\}_{n\in \mathbb{Z}} \in {\mathbb{N}_+}^ {\mathbb{Z}}$
 as the two-sided shift, i.e. $\h{\mathscr{G}} \{a_n\}_{n\in \mathbb{Z}}=  \{ a'_n \}_{n\in \mathbb{Z}}$ where $a'_n = a_{n+1}$. The map $\h{\mathscr{G}}$ admits the following geometric interpretation. Consider the domain $D(\h{\mathscr{G}}) = (0,1)\backslash {\mathbb{Q}} \times (0,1)\backslash \mathbb{Q}$. Let us identify the sequence $ \{a_n\}_{\mathbb{Z}}$ with the point  $\h{\alpha}  = (\h{\alpha} ^-, \h{\alpha}^+) \in D(\h{\mathscr{G}})$, which is given by
\begin{equation}\label{alpha+-}
\h{\alpha} ^+ = [a_1, a_2, \dots , a_n , \dots ]; \qquad \h{\alpha}^- = [a_0, a_{-1}, \dots , a_{-n} , \dots ].
\end{equation}
Then  $\h{\mathscr{G}}({\h{\alpha}})= \h{\beta}$ where  $\h{\beta}  = (\h{\beta} ^-, \h{\beta}^+)  $ and
\begin{equation*}
\h{\beta}^+ = \mathscr{G} ( \h{\alpha}^+ ) = \left\{ \frac{1}{\h{\alpha}^+}\right\} = \frac{1}{\h{\alpha}^+} - a_1; \qquad  \h{\beta}^- = \frac{1}{ \left[ \frac{1}{\h{\alpha}^+}\right]+ \h{\alpha}^- } = \frac{1}{ a_1 + \h{\alpha}^- }
\end{equation*}
Clearly, denoting by $\pi$ the projection $\pi( \h{\alpha}) = \h{\alpha}^+$ or equivalently $\pi( \{a_n\}_{n\in \mathbb{Z}}) =  \{a_n\}_{n\in \mathbb{N_+}}  $,  we have $\pi \h{\mathscr{G}}  =  \mathscr{G} \pi$.
The sequence $\{a_n\}_{n\in \mathbb{Z}}$ is the symbolic coding of $\h{\alpha} \in D(\h{\mathscr{G}})$ under $\h{\mathscr{G}}$ in the sense that
$a_n = \left[ \left( \pi \h{\mathscr{G}}^{n-1} \h{\alpha}\right)^{-1}  \right]$

The map $\h{\mathscr{G}}$ admits a natural invariant probability measure $\mu_2 $ on $D(\h{\mathscr{G}})$ which is given by the density:
\begin{equation*}
 \rho_2 (\alpha^-, \alpha^+) = \frac{1}{\ln 2(1+ \alpha^- \alpha^+)^2 }
\end{equation*}
\begin{rem}\label{mu2pimu1}
The Gauss measure $\mu_1$ can be recovered as $\pi_* \mu_2$, i.e. for each measurable set $A\subset (0,1)$, we have $\mu_1(A)= \mu_2(\pi^{-1}A)$.
\end{rem}

Given any  $\h{\alpha} = \{a_n\}_{n\in \mathbb{Z}}  \in D(\h{\mathscr{G}})$,  $\h{\alpha}^-$ and  $\h{\alpha}^+$ will always denote the two components of  $\h{\alpha} \in D(\h{\mathscr{G}})$ which are given explicitly in terms of the $a_n$ by (\ref{alpha+-}).

Let $q_n=q_n(\h{\alpha }) = q_n({\h{\alpha}^+ })$, $n\in \mathbb{N}_+$, be the sequence of denominators of the convergents of $\h{\alpha}^+ $. Also, given $R>0$, $n_R(\h{\alpha})$ and  $q_{n_R}(\h{\alpha})$ are set equal to the analogous quantities defined for $\alpha=\h{\alpha}^+ $.
\begin{rem}\label{n_Rfuture}
By construction, the functions $q_n$ (for any $n\in \mathbb{N_+}$),  $n_R$ and $q_{n_R}$ (for any $R>0$)  on $D(\h{\mathscr{G}})$
 are constant on fibers $\pi^{-1}\alpha$, $\alpha\in (0,1)\backslash {\mathbb{Q}}$.
\end{rem}

\subsection{Cylinders.}\label{cylinderssec}
For $b_k \in \mathbb{N}_+$, $k=1, \dots, n$, denote  $\Cp{b_1, \dots, b_n}$ the cylinder
\begin{equation*} \Cp{b_1, \dots, b_n} = \{ \alpha=\{a_n\}_{n\in \mathbb{N}_+} \in (0,1)\backslash{\mathbb{Q}} \, : \, a_k = b_k, 1\leq k\leq n \}.
 \end{equation*}
We will denote by $\mathscr{C}^+_n$ the set of all cylinders  of length $n$, i.e, the set of all $\Cp{b_1, \dots, b_n}$ with  $b_k\in \mathbb{N}_+$ for $1\leq k \leq n$. Moreover, if $\mathcal{C} \in \mathscr{C}^+_n$, we will denote by $\h{\mathcal{C} }$ the set $\pi^{-1}\mathcal{C} \subset D(\h{\mathscr{G}}) $.

More generally, given $b_k \in \mathbb{N}_+$, $k=0,\pm 1, \dots, \pm n$, let
\begin{equation*} \C{b_{-n},\dots, b_0;  b_1, \dots, b_n} = \{ \alpha=\{a_n\}_{n\in \mathbb{Z}} \in D(\h{\mathscr{G}}) \, : \, a_k = b_k, -n\leq k\leq n \}
 \end{equation*}
and $\mathscr{C}_n$ the set of all bi-sided cylinders  of length $n$, i.e, the set, as  $b_k\in \mathbb{N}_+$ for $-n \leq k \leq n$,
 of all $\C{b_{-n}, \dots,b_0; b_1, \dots,  b_n}$.
\begin{rem}\label{measureCn}
From the expression of the Gauss density and Remark \ref{mu2pimu1}, we get that $\mu_2(\hC{n})= \mu_1(\C{n})= \mu_1\left( \frac{1}{n+1}, \frac{1}{n} \right)=  O \left( \frac{1}{n^2}\right) $.
\end{rem}

\subsection{Special flows.}\label{suspensionflowsec}
Consider a probability space  $(D, \mathscr{B}, \mu_2)$ and an invertible map $F:D\rightarrow D$ which preserves $\mu_2$.
Let $\varphi: D\rightarrow \mathbb{R}_+$ be a strictly positive function such that $\int_{D} \varphi(\alpha) d\mu_2 < \infty$.
The {phase space $D_{\Phi}$} of the special flow is the subset of $D\times \mathbb{R}$ given by
\begin{equation*}
D_{\Phi} = \{ (x,y) |\quad  x \in D  \, : \,  0\leq y < \varphi(x) \}
\end{equation*}
and can be depicted as the set of points below the graph of the roof function $\varphi$.
Consider the normalized measure $\mu_3$ which is the restriction to $D_{\Phi}$ of the product measure $\left(\int_{D} \varphi(\alpha) d \mu_2 \right)^{-1} \mu_2 \times \lambda$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$.

The \emph{special flow $\{ \Phi_t \}_{t\in \mathbb{R}}$ built over $F$ with the help of the roof function $\varphi$} is a one-parameter group of $\mu_3$-measure preserving transformations of $D_\Phi$ whose action is generated by the following two relations:
\begin{equation} \label{suspflow}
 \left\{  \begin{array}{lll}
\Phi_t(x,y) &=& (x, y+t), \qquad \mathrm{if} \, 0\leq y+t < \varphi(x); \\
\Phi_{\varphi(x)}(x,0) &=& ( F(x),0).\\
		     \end{array} \right.
\end{equation}
Under the action of the flow a point of $(x,y) \in D_\Phi$  moves with unit velocity along the vertical line up to the point $(x,\varphi(x))$, then jumps instantly to the point $\left( F(x),0 \right)$, according to the base transformation. Afterward it continues its motion along the vertical line until the next jump and so on (see e.g. \cite{CFS:erg}).
Abusing the notation, we will often identify any set $C\subset D$ with $C\times \{ 0 \} \subset D\times \{0\}\subset D_{\Phi}$.

We will denote
by
\begin{equation*} \BS{\varphi,F}{0}( x) := 0; \qquad
\BS{\varphi,F}{r}( x) = \BS{\varphi}{r}( x)  := \sum_{i=0}^{r-1} \varphi(F^i(x)), \qquad x\in D, \, r\in \mathbb{N}^+,
\end{equation*}
the $r^{th}$ non-renormalized \emph{Birkhoff sum} of $\varphi$ along the trajectory of $x$ under $F$.

Let $t>0$. Given $x\in D$ denote by $r(x,t)$ the integer uniquely defined by
\begin{equation} \label{defr}
r(x,t):= \min \{ r\in \mathbb{N} \, | \quad  \BS{\varphi}{r}(x) > t \}.
\end{equation}
Then $r(x,t)-1$ gives the number of discrete iterations of $F$ which the point $(x,0)$ undergoes before time $t$.
According to this notation the flow $\Phi_t$ defined by (\ref{suspflow}) acts as
\begin{equation} \label{flowdef}
\Phi_t(x,0) = \left( F^{r(x,t)-1}(x), t- \BS{\varphi}{r(x,t)-1}(x)\right).
\end{equation}
For $t<0$, the action of the flow is defined as the inverse map.

\section{Reduction to a special flow.}\label{reductionsec}
Let us first define the special flow that we are going to consider.
\subsection{Roof function.}
Consider the following positive real-valued function on $ D(\h{\mathscr{G}})$:
\begin{equation} \label{varphidef}
\varphi(\h{\alpha}) = \ln \left( a_1 + \frac{1}{a_0 + \frac{1}{a_{-1} + {\dots }}} \right) = -\ln (\h{\mathscr{G}}\h{\alpha})^- .
\end{equation}
The reason for this definition will be clear after Lemma \ref{logqnandBS}.
It is easy to see, from  Remark \ref{measureCn}, that
the function $\varphi $ is integrable with respect to $\mu_2$.

\begin{rem}\label{infsup}
Let $\mathcal{C} \in \mathscr{C}_n$, $n\geq 1$, be any cylinder. Then there exists $\delta = \delta(\mathcal{C})>0$ and $M=M(\mathcal{C})>0$ such that $\inf_{\h{\alpha}\in \h{\mathcal{C}}} \varphi (\h{\alpha})\geq \delta$ and $\sup_{\h{\alpha}\in \h{\mathcal{C}}} \varphi (\h{\alpha})\leq M$. It follows by remarking that $\varphi(\h{\alpha}) \geq \ln (1+\frac{1}{a_0+1})$ and  $\varphi(\h{\alpha}) \leq \ln (a_1 + 1)$.
\end{rem}
Let us consider the special flow $\{ \Phi_t \}_{t\in \mathbb{R}}$ built over $\h{\mathscr{G}}$ under the function $\varphi$ and let $\mu_3 = (\int \varphi \mathrm{d} \mu_2)^{-1} \mu_2 \times \lambda$ be the  $\Phi_t$-invariant  probability  measure.
Let us recall that $\{ \Phi_t \}_{t\in \mathbb{R}}$ is said to be mixing if, for all Borel subsets $A, B$ of $D_{\Phi}$, we have
\begin{equation*}
\lim_{t\rightarrow \infty} \mu_3 \left( \Phi_{-t} (A) \cap B \right) = \mu_3(A)\mu_3(B).
\end{equation*}
\begin{prop}\label{mixing}
The flow $\{ \Phi_t \}_t $ is mixing.
\end{prop}
\noindent Proposition \ref{mixing} was proved in \cite{DS:sta}. We recall the proof in \S \ref{mixingsection}.

\subsection{Denominators growth and Birkhoff sums.}
 Let us show that $\ln q_n$ can be approximated by Birkhoff sums of $\varphi$.
Indeed, put
\begin{equation} \label{deffn}
f_n(\h{\alpha}) = \ln q_n (\h{\alpha}) - \BS{\varphi}{n} (\h{\alpha}).
\end{equation}
\begin{lemma}\label{logqnandBS}
There exist a function $f$ on $D(\h{\mathscr{G}})$ such that $f_n $ converges to $f$ uniformly in $\h{\alpha}$ and exponentially fast in $n$, i.e.
\begin{equation} \label{errorf}
\ln q_n (\h{\alpha}) = \BS{\varphi}{n} (\h{\alpha}) + f (\h{\alpha}) + \epsilon_n(\h{\alpha}), \qquad  \sup_{\h{\alpha}\in D(\h{\mathscr{G}})} \epsilon_n (\h{\alpha})  = O( 2^{- n}).
\end{equation}
\end{lemma}
\begin{proof}
Let  $q_0=1$, $q_{-1}=0$ and $r_n = \frac{q_n}{q_{n-1}}$ for $n\in \mathbb{N}_+$,  so that $q_n = \Pi_{k=1}^{n} r_k$. From the well-known relation  $q_{k+1} = a_{k+1} q_k + q_{k-1}$ which holds for $k\geq 0$ (see e.g. \cite{Kh:con}), we get by recursion that, for $k\geq 1$,
\begin{equation}
r_{k+1}= a_{k+1}+ \frac{1}{r_k} = [a_{k+1}; a_k , \dots, a_1]
\end{equation*}
and hence
\begin{equation} \label{logqn}
\log q_n = \sum_{k=1}^n \ln[a_{k}; a_{k-1} , \dots, a_1]  .
\end{equation}

Since $(\h{\mathscr{G}}^k \h{\alpha})^- = [a_k, a_{k-1}, \dots ]$, from the definition (\ref{varphidef}) of $\varphi$ we get
\begin{equation} \label{BS}
\BS{\varphi}{n}(\h{\alpha}) = \sum_{k=1}^n \ln \frac{1}{(\h{\mathscr{G}}^k \h{\alpha})^-} = \sum_{k=1}^{n}\ln [a_{k}; a_{k-1}, \dots ].
\end{equation}
Thus, from (\ref{deffn}) and (\ref{logqn}),
\begin{equation*}
({f}_{k+1} - {f}_{k} )(\h{\alpha}) =  \ln r_{k+1} - \varphi(\h{\mathscr{G}}^{k} \h{\alpha}) =  \ln r_{k+1} - \ln \frac{1}{(\h{\mathscr{G}}^{k+1} \h{\alpha})^-} = \ln \frac{[a_{k+1}; a_k, \dots, a_1 ]}{[a_{k+1}; a_k,\dots ]}.
\end{equation*}
In order to estimate the last term, consider $\beta_k = [ a_k, a_{k-1},\dots ]$ and and let $\left\{ \frac{p_{k,m}}{q_{k,m}}\right\}_m$ be the convergents of $\beta_k$, so that in particular $\frac{p_{k,k}}{q_{k,k}}=[ a_k, \dots, a_1 ] $. Recalling the well-known formula (see \cite{Kh:con})
\begin{equation}\label{bestapprox}
\left| \beta_k-\frac{p_{k,m}}{q_{k,m}} \right| \leq \frac{1}{(q_{k,m})^2}
\end{equation}
and using that for all sequences of denominators of convergents $q_{k,m} \geq 2 ^{\frac{m-1}{2}}$ (\cite{Kh:con}, Thm. 12) and that $[a_{k+1}; a_k,\dots ] \geq 1$, we get
\begin{equation} \label{exponential}
 \left|\ln
\frac{[a_{k+1}; a_k, \dots, a_1 ]}{[a_{k+1}; a_k,\dots ]}
 \right| \leq \left|\ln \left( 1 + \frac{ \frac{p_{k,k}}{q_{k,k}}- \beta_k}{[a_{k+1}; a_k, \dots ]} \right)  \right|\leq 2 \left| \beta_k-\frac{p_{k,k}}{q_{k,k}} \right| \leq 2^{2-k}.
\end{equation}
Hence, (\ref{exponential}) shows that we can well define
\begin{equation} \label{fdef}
 {f} (\h{\alpha}) =
\sum_{k=0}^{\infty} ({f}_{k+1} - {f}_{k} )(\h{\alpha})
\end{equation}
and (\ref{errorf}) clearly follows from the geometric bound of the series, with $\epsilon_n = 2^{3-n}$.
\end{proof}
\begin{lemma}\label{variationflemma}
If ${\alpha}_1,{\alpha}_2 \in \C{a_{-n},\dots, a_0; a_1, \dots , a_n}$, then
\begin{equation}  \label{variationf}
| f (\h{\alpha}_1) -   f (\h{\alpha}_2) |\leq C 2^{- n},
\end{equation}
where $C>0$ is an absolute constant.
\end{lemma}
\begin{proof}
First, by Lemma \ref{logqnandBS}, we have
\begin{equation} \label{difference}
|{f}(\h{\alpha}_1)- {f}(\h{\alpha}_2 )|\leq   C 2^{-n} +  |{f_n}(\h{\alpha}_1)- {f_n}(\h{\alpha}_2 )|.
\end{equation}
Let us estimate the second term of the right hand side.
Let $\h{ \alpha}_1  = \{a'_m\}_{m\in \mathbb{Z}}$, $\h{\alpha}_2 = \{a''_m\}_{m\in \mathbb{Z}}$, where by assumption $a'_m=a''_m=a_m$ for $-n\leq m\leq n$.
 From (\ref{BS}),
\begin{equation}\label{BSdifference}
  \BS{\varphi}{n}(\h{\alpha}_1)- \BS{\varphi}{n}(\h{\alpha}_2 ) = \sum_{k=1}^{n} \ln \frac{ [a_{k}; a_{k-1}, \dots, a_{-n}, a'_{-n-1}, a'_{-n-2} \dots  ]}{ [a_{k}; a_{k-1}, \dots, a_{-n}, a''_{-n-1}, a''_{-n-2} ,\dots ]}.
\end{equation}
Arguing as in Lemma \ref{logqnandBS}, for $k=1, \dots, n$, consider
\begin{equation*}
\beta'_k= [a_{k-1},a_{k-2}, \dots, a_{-n}, a'_{-n-1}, \dots] \quad \mathrm{and}\quad  \beta''_k= [a_{k-1},a_{k-2}, \dots, a_{-n}, a''_{-n-1}, \dots]
\end{equation*}
 and denote  by $\left\{\frac{ p^{'}_{k,n}}{q^{'}_{k,n}}\right\}_n$ and $\left\{ \frac{p^{''}_{k,n}}{q^{''}_{k,n}}\right\}_n$ their respective convergents. Since $\frac{p^{'}_{k,k+n}}{q^{'}_{k,k+n}} =\frac{p^{''}_{k,k+n}}{q^{''}_{k,k+n}}$,  from (\ref{bestapprox}) and $q_{m} \geq 2 ^{\frac{m-1}{2}}$ we get
\begin{equation*}
\left|\beta'_k - \beta''_k \right| \leq \left|\beta'_k - \frac{p^{'}_{k,k+n}}{q^{'}_{k,k+n}}\right| +  \left|\beta''_k - \frac{p^{''}_{k,k+n}}{q^{''}_{k,k+n}}\right| \leq \frac{1}{(q^{'}_{k,k+n})^2} + \frac{1}{{(q^{''}_{k,k+n})} ^2} \leq 2^{2-k-n}.
\end{equation*}
Hence, as in (\ref{exponential}),
\begin{equation*}
 \left| \sum_{k=1}^{n} \ln \frac{ [a_{k}; \dots, a_{-n}, a'_{-n-1}, \dots  ]}{ [a_{k}; \dots, a_{-n}, a''_{-n-1}, \dots ]}  \right| \leq   \sum_{k=1}^{n}
 2 \left| \beta'_k-\beta''_k \right| \leq  \sum_{k=1}^{n}  2^{3-k-n} \leq C 2^{-n}.
\end{equation*}
Hence,  by (\ref{deffn}), remarking that we also have $q_n(\h{ \alpha}_1^+)= q_n(\h{ \alpha}_2^+)$, this implies that $ |{f_n}(\h{\alpha}_1)- {f_n}(\h{\alpha}_2 )| \leq C 2^{-n}$ and gives the desired estimate of (\ref{difference}).
\end{proof}

\subsection{Comparing renewal times.
Given $\h{\alpha}$ and $R>0$, we want to choose  $T$ as a function of $R$ so that we can compare $n_R(\h{\alpha})$ and $r(\h{\alpha}, T)$, where $r(\h{\alpha}, T)-1$ is the discrete number of iterations undergone by $\Phi_t (\h{\alpha},0)$ for $t\leq T$, see (\ref{defr}).
Let us recall that $n_R(\h{\alpha})$ is uniquely determined by
\begin{equation*}
\ln q_{n_R(\h{\alpha})-1} \leq  \ln R < \ln q_{n_R(\h{\alpha})}.
\end{equation*}
By  (\ref{deffn}), the previous inequality can be rewritten as
\begin{equation} \label{n_Rinequality}
 \BS{\varphi}{n_R(\h{\alpha})-1} (\h{\alpha})+ f_{n_R(\h{\alpha})-1}(\h{\alpha}) \leq \ln R <  \BS{\varphi}{n_R(\h{\alpha})} (\h{\alpha}) + f_{n_R(\h{\alpha})}(\h{\alpha}).
\end{equation}
To avoid the dependence of the time on $\alpha$, let us localize to a set of $\mathcal{C} \subset D(\h{\mathscr{G}})$ and denote $f_{\mathcal{C}} = \sup_{\h{\alpha} \in \mathcal{C} } f(\h{\alpha})$. Assume that for all $\h{\alpha}\in \mathcal{C}$ we have $| f (\h{\alpha}) -  f_{\mathcal{C}}|\leq \epsilon/2$ (such sets will be constructed in the Proof of Theorem \ref{main}).

Let us first show that on large measure sets the growth of $n_R$ is guaranteed by the growth of $R$.
\begin{lemma}\label{badpoints}
For each measurable $\mathcal{C}\subset D(\h{\mathscr{G}})$ and $\epsilon>0$ there exits a measurable $ \mathcal{C} '\subset \mathcal{C}$
such that $\mu_2(\mathcal{C}\backslash \mathcal{C}')\leq \epsilon \mu_2(\mathcal{C})$ and $\min_{\h{\alpha}\in \mathcal{C}'}  n_R(\h{\alpha} ) $ tends to infinity uniformly as $R$ tends to infinity.

Similarly, given  $\epsilon>0$ there exits a measurable $ \mathcal{C}_{\epsilon} \subset (0,1)$
such that $\mu_1((0,1)\backslash \mathcal{C}_{\epsilon})\leq \epsilon $ and $\min_{{\alpha}\in \mathcal{C}_{\epsilon}}  n_R({\alpha} ) $ tends to infinity uniformly as $R$ tends to infinity.
\end{lemma}
\begin{proof}
By L{\'e}vy-Khinchin Theorem, for $\mu_1$-a.e. $\alpha \in (0,1)$, there exists an absolute constant $l>0$ such that $\lim_{n\rightarrow \infty} \frac{\ln q_n}{n} =  l$.
 By Remarks \ref{mu2pimu1} and \ref{n_Rfuture}, the same holds for  $\mu_2$-a.e. $\h{\alpha }\in D(\h{\mathscr{G}})$. By Egorov's theorem, we can find for each $\epsilon>0$ a measurable subset $\mathcal{C}_1\subset \mathcal{C}$ with $\mu_2(\mathcal{C}\backslash \mathcal{C}_1 )\leq \frac{\epsilon}{2} \mu_2(\mathcal{C})$ on which the convergence is uniform, so that for some $\overline{n}$, $\frac{\ln q_n}{n} \leq 2 l$ for each $n\geq \overline{n}$. Moreover, there exists $\mathcal{C}_2\subset \mathcal{C}$ with  $\mu_2(\mathcal{C}\backslash \mathcal{C}_2 )\leq \frac{\epsilon}{2} \mu_2(\mathcal{C})$ such that on $\mathcal{C}_2$, the functions $\frac{\ln q_n}{n} $ for $n=0,\dots, \overline{n}$ are uniformly bounded.  Hence, setting $\mathcal{C}'= \mathcal{C}_1\cap \mathcal{C}_2$, $\mu_2(\mathcal{C}\backslash \mathcal{C}' )\leq \epsilon \mu_2(\mathcal{C})$ and there exists a constant $c=c(\mathcal{C} , \epsilon)>0$ such that for all $\h{\alpha}\in \mathcal{C}' $ and all $n\in \mathbb{N}_+$ we have $\ln q_{n} \leq c n $. Since by definition $q_{n_R(\h{\alpha})}>R$, this implies that $\min_{\h{\alpha}\in \mathcal{C}'}  n_R(\h{\alpha}) \geq (c)^{-1} \ln q_{n_R(\h{\alpha})}  \geq (c)^{-1} \ln R $, from which the Lemma follows.
The proof of the second part proceeds in exactly the same way.
\end{proof}

\begin{lemma} \label{comparing}
Assume that $\mathcal{C}\in \mathscr{C}_n$ and for all $\h{\alpha}\in \mathcal{C}$ we have $| f (\h{\alpha}) -  f_{\mathcal{C}}|\leq \epsilon/2$. There exists $R_0 =R_0(\mathcal{C}) >0$ such that, whenever $R\geq R_0$, if we set $T=T(R, \mathcal{C})=  \ln R - f_{\mathcal{C}}$ and consider
\begin{equation*}
U = U_{\mathcal{C}}: =\{\h{\alpha}  \in \mathcal{C}
\, : \,  n_R(\h{\alpha}) \neq r(\h{\alpha}, T) \},
\end{equation*}
 we have $\mu_2(U \cap \mathcal{C})\leq 7 \epsilon \mu_2(\mathcal{C})$.
\end{lemma}
\noindent Hence, outside of a subset of $\mathcal{C}$ of arbitrarily small proportion and for large $R$, the function $T=T(R,\mathcal{C})$ is such that $ n_R(\cdot) = r(\cdot, T)$.
\begin{proof}
 Let $T=T(R, \mathcal{C})=  \ln R - f_{\mathcal{C}}$.

By definition, we have
\begin{equation} \label{r(T)inequality}
 \BS{\varphi}{r(\h{\alpha}, T)-1} (\h{\alpha})\leq  T = \ln R - f_{\mathcal{C}}  <  \BS{\varphi}{r(\h{\alpha}, T)} (\h{\alpha}).
\end{equation}
Let $\mathcal{C}'\subset \mathcal{C}$ be given by Lemma \ref{badpoints}.
Let us show, by comparing (\ref{r(T)inequality}) to (\ref{n_Rinequality}), that,  as long as $R\geq R_0$ for some $R_0$ defined below, we have
\begin{equation}\label{inclusion}
U \cap \mathcal{C} ' \subset U_{\epsilon} \cup U_{-\epsilon} \cap \mathcal{C} '
\end{equation}
 where  $U_{\pm \epsilon}\subset D(\h{\mathscr{G}})$ are defined as
\begin{equation} \label{Uepsilondef}
 U_{-\epsilon} = \{ \h{\alpha}
  \, : \,  T < \BS{\varphi}{r(\h{\alpha}, T)}(\h{\alpha}) \leq T + \epsilon \}; \quad  U_{\epsilon} = \{ \h{\alpha}
 \, : \,  T - \epsilon < \BS{\varphi}{r(\h{\alpha}, T)-1}(\h{\alpha}) \leq T \}.
\end{equation}
To show the inclusion (\ref{inclusion}), assume that $ \h{\alpha}\in \mathcal{C} '$, but $ \h{\alpha}\notin U_{\epsilon} \cup U_{-\epsilon}$. Then, by (\ref{Uepsilondef}) and (\ref{r(T)inequality}), we have
\begin{equation} \label{strictr(T)inequalities}
 \BS{\varphi}{r(\h{\alpha}, T)-1} (\h{\alpha})\leq T-\epsilon; \qquad \BS{\varphi}{r(\h{\alpha}, T)}(\h{\alpha}) > T + \epsilon .
\end{equation}
Choose $n_0\gg 1$ so that by Lemma \ref{logqnandBS}, for all $n\geq n_0-1$, $|f_n - f|\leq \epsilon/2$. By {Lemma \ref{badpoints}} we can choose $R_0$ so that, if $R\geq R_0$, then $n_R(\h{\alpha})\geq n_0$ for all $\h{\alpha}\in \mathcal{C}'$.
 This, together with the  assumptions on $\mathcal{C}$, implies that, since $\h{\alpha}\in \mathcal{C}'$,   $|f_{i}(\h{\alpha}) -  f_{\mathcal{C}}|<\epsilon$ for $i=n_R(\h{\alpha}), i=n_R(\h{\alpha})-1$. Hence, (\ref{n_Rinequality}) gives
\begin{equation*}
 \BS{\varphi}{n_R(\h{\alpha})}  > \ln R -  f_{n_R(\h{\alpha})}(\h{\alpha}) > T -\epsilon; \quad
 \BS{\varphi}{n_R(\h{\alpha})-1} (\h{\alpha}) \leq \ln R - f_{n_R(\h{\alpha})-1}(\h{\alpha})  < T+ \epsilon,
\end{equation*}
which, compared with (\ref{strictr(T)inequalities}), since  $\BS{\varphi}{n}(\h{\alpha})$ is increasing in $n$, implies that
\begin{equation*}
r(\h{\alpha}, T)-1< n_R(\h{\alpha}) \quad \mathrm{and} \quad n_R(\h{\alpha})- 1 < r(\h{\alpha}, T) .
\end{equation*}
Since both $n_R(\h{\alpha})$ and $r(\h{\alpha}, T)$ are integers, these inequalities imply $n_R(\h{\alpha})=r(\h{\alpha}, T)$, i.e, $\h{\alpha}\notin U$, proving (\ref{inclusion}) as desired.

Recalling the definition (\ref{flowdef}) of the special flow action, we can rewrite the sets $U_{\pm\epsilon}$ as
\begin{eqnarray}\nonumber
 U_{\epsilon} &= &\{ \h{\alpha}
 \, : \,  0 \leq T- \BS{\varphi}{r(\h{\alpha}, T)-1} < \epsilon \} = \{ (\h{\alpha}, 0)
 \, : \,  \Phi_{T}\left(\h{\alpha}, 0\right) \in D_{\Phi}^{\epsilon} \} ,\\
 U_{-\epsilon} &= &\{ \h{\alpha}
\, : \,   \varphi(\h{\mathscr{G}}^{r(\h{\alpha}, T)-1}\h{\alpha})-\epsilon \leq  T- \BS{\varphi}{r(\h{\alpha}, T)-1} <   \varphi(\h{\mathscr{G}}^{r(\h{\alpha}, T)-1}\h{\alpha}) \} = \nonumber \\ &=&\{ (\h{\alpha}, 0)
\, : \,  \Phi_{T}\left(\h{\alpha}, 0\right)  \in  D_{\Phi}^{-\epsilon}  \},
\nonumber
\end{eqnarray}
where $D_{\Phi}^{\epsilon} = D(\h{\mathscr{G}})\times [0, \epsilon)$ and $D_{\Phi}^{-\epsilon} = \{ (\h{\alpha}, y) \, : \,  \varphi(\h{\alpha})-\epsilon \leq y <  \varphi(\h{\alpha})\}$.

We want to use mixing of $\{\Phi_t\}_{t\in \mathbb{R}}$ to estimate the measures of the last two sets. In order to do this, we need to ``thicken'' them as follows. Choose  $0< \delta \leq \epsilon$ such that, by Remark \ref{infsup}, $\delta < \min_{\h{\alpha}\in \mathcal{C}} \varphi(\h{\alpha})$
and  consider the following two subsets of $D_{\Phi}$:
\begin{equation*}
 U_{\pm\epsilon}^\delta = \{ (\h{\alpha}, z) \, : \,  0\leq z < \delta;\,  \Phi_{T}\left(\h{\alpha}, z\right) \in D_{\Phi}^{\pm \epsilon } \}=  D_{\Phi}^{\delta }  \cap  \Phi_{-T}  D_{\Phi}^{\pm \epsilon } .
\end{equation*}
Let us show that if  $\h{\alpha}\in U_{\epsilon}\cap \mathcal{C}$, then for each $0\leq z < \delta$, we have $(\h{\alpha}, z) \in  U_{\epsilon+\delta}^\delta $. Indeed, by choice of $\delta$,
\begin{equation}\label{cases} \begin{split}
& \Phi_T(\h{\alpha},z) = \Phi_{T+z}(\h{\alpha},0) = \Phi_{z} (\h{\mathscr{G}}^{r(\h{\alpha}, T)-1} \h{\alpha}, T - \BS{\varphi}{r(\h{\alpha}, T)-1} (\h{\alpha}))=\\&=\left\{\begin{array}{l} \left( \h{\mathscr{G}}^{r(\h{\alpha}, T)-1} \h{\alpha}, T+z  - \BS{\varphi}{r(\h{\alpha}, T)-1} (\h{\alpha})\right) \quad
\left\{ \begin{array}{l}\mathrm{if} \,\, \BS{\varphi}{r(\h{\alpha}, T)}(\h{\alpha}) > T+z ;
\\ \mathrm{and}  \, r(\h{\alpha}, T)= r( \h{\alpha}, T+z);  \end{array} \right. \\
\left(\h{\mathscr{G}} ^{r-1} \h{\alpha} , T+z  - \BS{\varphi}{r-1} (\h{\alpha}) \right)
\quad
\left\{ \begin{array}{l} \mathrm{if} \,\, \BS{\varphi}{r(\h{\alpha}, T)}(\h{\alpha}) \leq  T+z
\\ \mathrm{and} \, r=r(\h{\alpha}, T+z ) >  r(\h{\alpha}, T) .\end{array} \right.
\end{array} \right.
     \end{split}
\end{equation}
Since $r> r(\h{\alpha}, T)$ and $\BS{\varphi}{r}$ is increasing in $r$ and  by definition of $U_{\epsilon}$, both  $ T+z  - \BS{\varphi}{r-1} \leq T+\delta  - \BS{\varphi}{r(\h{\alpha}, T)-1} (\h{\alpha})\leq \epsilon + \delta$, hence  $(\h{\alpha}, z) \in  U_{\epsilon+\delta}^\delta  $.

Reasoning in a similar way, let us also show that if  $\h{\alpha}\in U_{-\epsilon}\cap \mathcal{C}$, then for each $0\leq z < \delta$, we have $(\h{\alpha}, z) \in   U_{-\epsilon}^\delta \cup U_{\delta}^\delta $. Indeed, from (\ref{cases}), in the first case $T+z -\BS{\varphi}{r(\h{\alpha}, T)-1} < \varphi(\h{\mathscr{G}}^{r(\h{\alpha}, T)-1}) (\h{\alpha}) $ and since  $\h{\alpha}\in U_{-\epsilon}$, also $T+z -\BS{\varphi}{r(\h{\alpha}, T)-1}  \geq T -\BS{\varphi}{r(\h{\alpha}, T)-1} \geq \varphi(\h{\mathscr{G}}^{r(\h{\alpha}, T)-1}) (\h{\alpha})  -\epsilon $, so    $(\h{\alpha}, z) \in   U_{-\epsilon}$;  while in the second case, since $r-1 \geq r(\h{\alpha}, T)$, we have $\BS{\varphi}{r}(\h{\alpha})\geq \BS{\varphi}{r(\h{\alpha}, T)}(\h{\alpha}) > T$ and hence  $0\leq T+z  -\BS{\varphi}{r}  < z < \delta$, so  $(\h{\alpha}, z) \in   U_{\delta}$.

Let $\mathcal{C}'_{\delta} = \mathcal{C}' \times [0,\delta)$. Hence, recalling also (\ref{inclusion}), we proved that
\begin{equation*}
(U\cap \mathcal{C}') \times [0,\delta) \subset ((U_{\epsilon} \cup U_{-\epsilon })\cap \mathcal{C}' )\times [0,\delta) \subset \mathcal{C}'_\delta \cap  U_{-\epsilon}^\delta \cup U_{\delta+\epsilon}^\delta = \mathcal{C}'_{\delta}\cap \Phi_{-T} (D_{\Phi}^{-\epsilon} \cup D_{\Phi}^{\delta+\epsilon} ) .
\end{equation*}
Considering the measures of the above sets and using mixing (Proposition \ref{mixing}), one can find $T_0$ such that, as soon as $T\geq T_0$,
 we have
\begin{equation}\label{mixingcons}
 \begin{split}
\mu_2(U\cap \mathcal{C}')\delta  =
\mu_3 (U\cap \mathcal{C}' \times [0,\delta) )& \leq \mu_3 (\mathcal{C}'_{\delta}\cap \Phi_{-T} (D_{\Phi}^{-\epsilon} \cup D_{\Phi}^{\delta+\epsilon} ) ) \leq  \\ &
\leq 2\mu_2(\mathcal{C}')\delta \mu_3( D_{\Phi}^{-\epsilon} \cup D_{\Phi}^{\delta+\epsilon}  )  \leq 2 \mu_2(\mathcal{C}') \delta (3\epsilon)
,    \end{split}
\end{equation}
where the last inequality follows from the fact that $\mu_3( D_{\Phi}^{\pm\epsilon} )\leq \epsilon$ and gives $\mu_2(U\cap \mathcal{C}') \leq 6\epsilon \mu_2(\mathcal{C}')$. Enlarging $R_0$ if necessary so that $\ln R_0 -f_{\mathcal{C}}\geq T_0$, if $R\geq R_0$ also $T\geq T_0$ and (\ref{mixingcons}) holds. Hence
 $\mu_2(U\cap \mathcal{C})\leq  \mu_2(U\cap \mathcal{C}\backslash \mathcal{C}') + 6\epsilon  \mu_2(\mathcal{C}) \leq 7\epsilon \mu_2(\mathcal{C})$, concluding the proof of the Lemma.
\end{proof}

\section{Existence of the limiting distribution.}\label{limitingsec}
\begin{proofof}{Theorem}{main}
Assume $b> a >1$ and $c_k\in \mathbb{N}_+$, $0\leq k< N$. We want to estimate the expression (\ref{limiting}).
Recalling Remark \ref{n_Rfuture}, as soon as $n_R({\alpha})>N$
we can rewrite the condition $ a_{n_R({\alpha})-k} = c_k, \, 0\leq k < N $ in an equivalent way as
\begin{equation}\label{conditionona}
\h{\mathscr{G}}^{n_R(\h{\alpha})-1}\h{\alpha} \in C_N, \qquad \mathrm{where} \quad \h{\alpha}^+=\alpha, \quad C_N := \h{\mathscr{G}}^{N-1} \hC{c_{N-1,} c_{N-2} \dots, c_0 },
\end{equation}
since if (\ref{conditionona}) holds,  $\{ a'_j \}_{j\in \mathbb{Z}} :=  \h{\mathscr{G}}^{n_R(\h{\alpha})-N} \h{\alpha} = \h{\mathscr{G}}^{-(N-1)} \h{\mathscr{G}}^{n_R(\alpha)-1}\h{\alpha}\in \hC{ c_{N-1} \dots, c_0 } $, so  $a_k'= c_{N-k}$ for $1\leq j \leq N$ and  $a'_j = a_{n_R(\alpha)-N+ j}$ by definition of $\h{\mathscr{G}}$,
hence $a_{n_R(\alpha)-N+ j} = c_{N-k}$ for $1\leq j \leq N$ gives the desired set of equalities for $k=N-l$.

Given two functions $g_1, g_2$ on $D(\h{\mathscr{G}})$, $g_1\leq g_2$, let us denote by $D_{\Phi}(g_1,g_2)$ the following subsets:
\begin{equation*}
D_{\Phi}(g_1,g_2) = \{ (\h{\alpha}, y) \in D_{\Phi} \, : \,  \varphi(\h{\alpha}) - g_2(\h{\alpha}) < y < \varphi(\h{\alpha}) - g_1 (\h{\alpha}) \}.
\end{equation*}
Notice that for some values of $g_1( \h{\alpha})$, $g_2( \h{\alpha})$, the  corresponding set of $y$ can be empty. Also, let $p(x,y)=x$ be the projection to the base of the special flow.
\begin{rem}\label{inclusionsDgs}
If $g_1'\leq g_1$ and $g_2'\geq g_2$, then $D_{\Phi}(g_1,g_2)\subset D_{\Phi}(g_1',g_2')$.
\end{rem}

We will show that the limiting distribution (\ref{limiting}) exists and is given by
\begin{equation}\label{limitingdistribution}
P_N \left( (a,b) \times \{ c_0\} \times \dots \times \{ c_{N-1}\} \right) = \mu_3 \left(  D_{\Phi}(\ln a,\ln b)  \cap p^{-1} C_N \right).
\end{equation}
Take  $\epsilon>0$.
For each $n\in \mathbb{N}$, the cylinders $\{ {\mathcal{C}} \, : \,  \mathcal{C}\in \mathscr{C}_n \}$, constitute a countable partition of $D(\h{\mathscr{G}})$.
Choose  $n$  so large that, by  Lemma \ref{variationflemma}, we have $| f (\h{\alpha}_1) -   f (\h{\alpha}_2) |\leq \epsilon/2$ for all $\h{\alpha}_1, \h{\alpha}_2 \in \mathcal{C}$. Let
\begin{equation*}
A_{\mathcal{C} } := \left\{ \h{\alpha} \in {\mathcal{C}} \, : \,    a < \frac{q_{n_R(\h{\alpha})}(\h{\alpha} )}{R} < b, \quad \h{\mathscr{G}}^{n_R(\h{\alpha})-1}\h{\alpha} \in C_N \right\}.
\end{equation*}
By the second part of Lemma \ref{badpoints}, there exist $R_1>0$ such that if $R\geq R_1$ we have  $n_R(\alpha) > N $ for any $\alpha$ outside a set of $\mu_1$-measure less than $\epsilon$. Hence, by (\ref{conditionona}) (and Remarks \ref{mu2pimu1} and \ref{n_Rfuture}), we have

\begin{equation*}
 \left| \mu_1  \left( \alpha  \, : \,  a < \frac{q_{n_R(\alpha)}}{R} < b,  \quad a_{n_R(\alpha)-k} = c_k, \, 1\leq k \leq N  \right)
\sum_{\mathcal{C}\in \mathscr{C}_n} \mu_2( A_{\mathcal{C} }) \right| \leq 2\epsilon
\end{equation*}
Let us first reduce to a finite sum. Consider the finite subset  $\mathscr{C}_n^m$ of cylinders $\mathcal{C}=\C{a_{-n},\dots ; \dots, a_n}$ such that $a_i< m$ for $-n\leq i\leq n$. Since, if $\mathcal{C} \in \mathscr{C}_n\backslash \mathscr{C}_n^m$, there exists $-n\leq i\leq n$ and $k\geq m$ such that $a_i =k$, we have,  using Remark \ref{measureCn} and invariance of $\mu_2$,
\begin{equation} \label{remindersum}
\sum_{\mathcal{C}\in \mathscr{C}_n \backslash \mathscr{C}_n^m} \mu_2({\mathcal{C}}  ) \leq   \sum_{i=-n}^{n} \sum_{k=m}^{\infty} \mu_2 \left( \h{\mathscr{G}}^{i} \hC{k} \right) \leq (2n+1)  \sum_{k=m}^{\infty} O\left(\frac{1}{k^2}\right) = O\left(\frac{1}{m}\right).
\end{equation}
Hence, choosing $m$ large enough, we can make (\ref{remindersum}) less than $\epsilon$. To each $\mathcal{C} \in \mathscr{C}_n^m$ we can apply Lemma \ref{comparing} and hence, for $R\geq \max_{\mathcal{C} \in \mathscr{C}_n^m} R_0 (\mathcal{C})$ (where $R_0(\mathcal{C})$ and $U_{\mathcal{C}}$ are as in Lemma \ref{comparing}) we get
\begin{eqnarray}&& \label{partition}
\left|  \mu_1 \left( \alpha  \, : \,  a < \frac{q_{n_R(\alpha)}}{R} < b,  \quad a_{n_R(\alpha)-k} = c_k, \, 1\leq k \leq N  \right) - \sum_{\mathcal{C}\in \mathscr{C}_n^m } \mu_2 (A_{{\mathcal{C}} \backslash U_{\mathcal{C}} })\right| \leq
\nonumber \\&&\nonumber
\leq 2\epsilon + \left| \sum_{\mathcal{C}\in \mathscr{C}_n \backslash \mathscr{C}_n^m} \mu_2 (A_{\mathcal{C}} ) + \sum_{\mathcal{C}\in  \mathscr{C}_n^m} \mu_2 (A_{\mathcal{C} \cap U_\mathcal{C} }) \right| \leq 3 \epsilon + 7\epsilon
 \sum_{\mathcal{C}\in \mathscr{C}_n^m} \mu_2 ({\mathcal{C}} ) \leq  10 \epsilon ,
\end{eqnarray}
where the inequality  before the last follows from the observation that $A_\mathcal{C}\subset \mathcal{C}$, (\ref{remindersum})  and Lemma \ref{comparing}.

To conclude the proof and to get (\ref{limitingdistribution}), it is enough to prove that, for each $\mathcal{C}\in \mathscr{C}_n^m$, as long as $R$ is sufficiently large, we have
\begin{equation} \label{conditional}
\left|  \frac{\mu_2 (A_{\mathcal{C} \backslash U_{\mathcal{C}} } ) }{\mu_2 ({\mathcal{C} \backslash U_{\mathcal{C}} } )}
 -   \mu_3 \left(   D_{\Phi}(\ln a,\ln b) \cap p^{-1} C_N \right) \right| \leq \epsilon .
\end{equation}
Fix $\mathcal{C} \in \mathscr{C}_n^m $ and consider on $\mathcal{C}$ the function  $T= T(R) = \ln R - f_{{\mathcal{C}}} $ (recall that $f_{{\mathcal{C}}}= \sup_{{\mathcal{C}}} f $) and  $U=U_{\mathcal{C}}$ given by Lemma \ref{comparing}.
Since  by Lemma \ref{comparing},  for $R\geq R_0(\mathcal{C})$, on $\mathcal{C}\backslash U$ we have $n_R(\h{\alpha})=r(\h{\alpha}, T)$, applying (\ref{errorf}) of Lemma \ref{logqnandBS}, we get
\begin{equation*} \begin{split} &
 \left\{ \h{\alpha} \in \mathcal{C} \backslash U \, : \,  a < \frac{q_{n_R(\h{\alpha})}}{R} < b \right\} =  \{ \h{\alpha} \in {\mathcal{C}} \backslash U \, : \,  \ln a < \ln q_{r(\h{\alpha}, T)}(\h{\alpha})-\ln R < \ln b  \} = \\ & = \{ \h{\alpha} \in {\mathcal{C}} \backslash U  \, : \,  \ln a  <  \BS{\varphi}{r(\h{\alpha}, T)} (\h{\alpha}) -T + \epsilon_{R,\mathcal{C}}(\h{\alpha}) < \ln b \},
     \end{split}
\end{equation*}
where we denoted by $\epsilon_{R,\mathcal{C}}(\h{\alpha})= \epsilon_{n_R(\h{\alpha})}(\h{\alpha}) - f_{\mathcal{C}} + f(\h{\alpha}) $. Let us show that $| \epsilon_{R,\mathcal{C}}|\leq 2 \epsilon$  uniformly on  $ \mathcal{C} \backslash U $; indeed, by construction of $\mathcal{C}$, $ | f(\h{\alpha}) - f_{\mathcal{C}} |\leq \epsilon$;  moreover, since by (\ref{r(T)inequality}) and Remark \ref{infsup}, $\ln R - f_{\mathcal{C}}  <  \BS{\varphi}{r(\h{\alpha}, T)} (\h{\alpha}) \leq M(\mathcal{C})r(\h{\alpha}, T)  $, on  $\mathcal{C}\backslash U$ we have $n_R(\h{\alpha})=r(\h{\alpha}, T)\geq (\ln R - f_{\mathcal{C}})/M(\mathcal{C})$, so enlarging $R_0(\mathcal{C})$ we can assure that for each $\h{\alpha}\in \mathcal{C}\backslash U$ and $R\geq R_0$,  $n_R(\h{\alpha})$ is so large that, by Lemma \ref{logqnandBS}, also  $ \epsilon_{n_R(\h{\alpha})}(\h{\alpha})\leq \epsilon$.

Let us denote by $(\Phi_t(x,y))^v$ the vertical component $y'$ of $\Phi_t(x,y)=(x',y')$.
Using the definition of the flow action  (\ref{flowdef}) and the equality $n_R(\h{\alpha})=r(\h{\alpha}, T)$,
 we can rewrite
\begin{equation} \label{distance}
 \BS{\varphi}{r(\h{\alpha}, T)} (\h{\alpha}) -T = \varphi(\h{\mathscr{G}}^{n_R(\alpha)-1}\h{\alpha}) -\left( \Phi_{T} (\h{\alpha},0)\right)^v .
\end{equation}
Remark that $\h{\mathscr{G}}^{n_R(\h{\alpha})-1}\h{\alpha} = p (\Phi_{T} (\h{\alpha},0) )$ and that
condition  (\ref{conditionona}) can be expressed as $p (\Phi_{T} (\h{\alpha},0) )\in C_N$.
The quantity (\ref{distance}) represents geometrically the vertical distance of $\Phi_{T} (\h{\alpha},0)$ from the roof function

Hence, recalling the definitions of the sets $D_{\Phi}(g_1,g_2)$ and $C_N$ given at the beginning of the proof,
  we have shown that
\begin{equation*}
A_{\mathcal{C}\backslash U } :=

{\mathcal{C}} \backslash U \times \{0\}
\, \cap \, \Phi_{-T} \left( D_{\Phi}(g_1, g_2) \cap p^{-1}{C}_N \right), \qquad g_1 = \ln a -\epsilon_{R,\mathcal{C}}, \quad g_2= \ln b -\epsilon_{R,\mathcal{C}}.

\end{equation*}
Let us ``thicken'' the sets in order to apply mixing of $\{\Phi_t\}_{t\in \mathbb{R}}$ as in the proof of Lemma \ref{comparing}.
 If we choose $0< \delta < \min \{ \min_{\h{\alpha}\in \mathcal{C}} \varphi(\h{\alpha}), \epsilon\} $ (well defined by Remark \ref{infsup}), for each $ \h{\alpha} \in A_{\mathcal{C}\backslash U}$ and $0\leq z < \delta$, reasoning as in the proof of Lemma \ref{comparing} (see e.g. (\ref{cases})), we get  $(\h{\alpha} ,z ) \in  \Phi_{-T} \left(  D_{\Phi}(g_1-\delta , g_2) \cap p^{-1}C_N  \cup  D_{\Phi}^{\delta}\right) $. Hence, using also Remark \ref{inclusionsDgs} combined with  $\delta\leq \epsilon $ and  $|\epsilon_{R,\mathcal{C}} |\leq 2\epsilon$,
\begin{equation*}
\delta \mu_2(A_{\mathcal{C}\backslash U})
 \leq  \mu_3  \left( {\mathcal{C}} \backslash U \times [0,\delta) \cap  \Phi_{-T} \left(  D_{\Phi}(\ln a + 3\epsilon , \ln b - 2 \epsilon) \cap p^{-1}C_N  \cup  D_{\Phi}^{\delta}\right) \right)
\end{equation*}
Remark that $\mu_3({D_{\Phi}^\delta})\leq \epsilon$. Enlarging $R_0$, so that  if $R\geq R_0$ also $T(R)$ is sufficiently large, we can use  mixing (see Proposition \ref{mixing}) to ge
\begin{equation} \label{upper}
\delta \mu_2(A_{\mathcal{C}\backslash U}) \leq \delta \mu_2 ( {\mathcal{C}} \backslash U ) \left( \mu_3 \left(  D_{\Phi}(\ln a + 3\epsilon , \ln b - 2 \epsilon) \cap p^{-1}C_N \right) + 2\epsilon \right) .\end{equation}
In order to get the opposite inequality, one can show,  reasoning again as in the proof of Lemma \ref{comparing},  that if $(\h{\alpha} ,z ) \in  {\mathcal{C}} \backslash U \times [0,\delta)$ is such that $(\h{\alpha} ,z ) \in \Phi_{-T} \left(  D_{\Phi}(g_1 , g_2-\delta ) \cap p^{-1}C_N  \backslash  D_{\Phi}^{\delta}\right) $, we have that $\h{\alpha} \in A_{\mathcal{C}\backslash U}$. This means, using again also Remark \ref{inclusionsDgs}, that
\begin{equation*}
  {\mathcal{C}} \backslash U \times [0,\delta) \cap \Phi_{-T} \left(  D_{\Phi}(\ln a -3\epsilon , \ln b + 2 \epsilon) \cap p^{-1}C_N  \backslash  D_{\Phi}^{\delta}\right) \subset A_{\mathcal{C}\backslash U} \times [0,\delta) .
\end{equation*}
Applying again mixing, enlarging again $R_0$ if necessary, for $R\geq R_0$, (and using that for any measurable $D\subset D_{\Phi}$, we have $\mu_3\left( D\backslash  D_{\Phi}^{\delta}\right) \geq \mu_3(D) - \epsilon$) we get
\begin{equation} \label{lower}
\delta \mu_2(A_{\mathcal{C}\backslash U}) \geq \delta  \mu_2 ( {\mathcal{C}} \backslash U ) \left( \mu_3  \left(  D_{\Phi}(\ln a -3\epsilon , \ln b + 2 \epsilon) \cap p^{-1}C_N \right)
- 2 \epsilon
\right).
\end{equation}
Since moreover, by Fubini theorem,
\begin{equation*} \left| \mu_3 \left(  D_{\Phi}(\ln a \pm 3 \epsilon , \ln b \mp 2 \epsilon) \cap p^{-1}C_N \right) - \mu_3 \left(  D_{\Phi}(\ln a  , \ln b ) \cap p^{-1}C_N \right) \right| \leq 5 \epsilon ,
\end{equation*}
combining  (\ref{upper}) and (\ref{lower}) we get (\ref{conditional}) and hence conclude the proof of the existence of the limiting distribution.
\end{proofof}

\section{Mixing of the special flow.} \label{mixingsection}
In what follows we briefly outline the proof of Proposition \ref{mixing} given in \cite{DS:sta}.
\begin{proofof}{Proposition}{mixing}
Given a point $(\h{\alpha}_0, y_0)\in D_{\Phi}$, let us construct the local stable and unstable leaves through it, denoted by $\Gamma^{(s)}_{loc}(\h{\alpha}_0, y_0)$ and $\Gamma^{(u)}_{loc}(\h{\alpha}_0, y_0)$ respectively (as a general reference, see e.g. \cite{Si:top}).

Since the roof function $\varphi(\h{\alpha})$ depends only on $(\h{\mathscr{G}}\h{\alpha})^-$, it is easy to construct the local unstable leaf, which is given by a piece of a ``horizontal'' segment:
\begin{equation}\label{locstab}
\Gamma^{(u)}_{loc}(\h{\alpha}_0, y_0) \subset \{ (\h{\alpha}, y) \, : \,  \h{\alpha}^-=\h{\alpha}_0^-, \,  y=y_0 \}.
\end{equation}
The local stable leaf through $(\h{\alpha}_0, y_0)$ is given locally by the following curve parametrized by $\alpha^-$:
\begin{equation} \label{locunst}
\Gamma^{(s)}_{loc}(\h{\alpha}_0, y_0) \subset \left\{ (\h{\alpha}, y) \, : \,  \h{\alpha}^+=\h{\alpha}_0^+, \,  y= y_0 + \ln \frac{1+ \h{\alpha}^- \h{\alpha}^+_0}{1+ \h{\alpha}_0^- \h{\alpha}^+_0} \right\}.
\end{equation}
In order to see it, one can construct it as follows.
Let us denote by $(\h{\alpha}_t, y_t) = \Phi_t (\h{\alpha}_0, y_0)$. Consider a small ``vertical'' segment at  $(\h{\alpha}_t, y_t)$, i.e.
\begin{equation*}
\Gamma^t_{\delta_t}  = \{ (\h{\alpha}, y_t) \, : \,  \h{\alpha}^+=\h{\alpha}_t^+, \,  |\h{\alpha}^- - \h{\alpha}_t^-|< \delta_t \},
\end{equation*}
where $\delta_t$ is chosen sufficiently small so that, for some $\delta>0$,
\begin{equation*}
\Phi_{-t} (\Gamma^t_{\delta_t}  ) \subset \{ (\h{\alpha}, y) \, : \,  \h{\alpha}^+=\h{\alpha}_0^+,\,  | y-y_0| < \delta,  \, 0< y_0 -\delta < y < \varphi(\h{\alpha_0}) -\delta  \}.
\end{equation*}
Then, if $(\h{\alpha} , y)\in \Phi^{-t} (\Gamma^t_{\delta_t}  )$, by definition of special flow, since $r(t)(\h{\alpha}) = r(t)(\h{\alpha_0}) = r(t)$ by construction, we have $y - \BS{\varphi}{r(t)}(\h{\alpha}) = t = y_0 - \BS{\varphi}{r(t)}(\h{\alpha}_0)$.
Denote $\alpha_0^+= \{a^0_k\}_{k\in \mathbb{N}_+}$ and ${p_n}/{q_n}$
 its convergents. Let
\begin{equation}\label{betas}
\beta':=[a^0_1 + \h{\alpha}^-_0, a^0_2, \dots, a^0_{r(t)} ]=  \frac{1}{\h{\alpha}^-_0 + \frac{q_{r(t)}}{p_{r(t)}}}, \, \, \beta'':=[a^0_1 + \h{\alpha}^-, a^0_2, \dots, a^0_{r(t)} ]= \frac{1}{\h{\alpha}^- + \frac{q_{r(t)}}{p_{r(t)}}}
\end{equation}
and
${p'_n}/{q'_n}$ and ${p''_n}/{q''_n}$
their respective convergents.  Remark that $p'_{n }= p''_{n} = p_{n} $ for  $1\leq n\leq r(t)$ since they satisfy the same recursive equations  $p_{k+1} = a_{k+1} p_k + p_{k-1}$  for $2\leq k+1\leq r(t)$ with  initial data $p_0=0$, $p_1=1$. Hence, $\beta'= \frac{p_{r(t)}}{q'_{r(t)}}$ and  $\beta''= \frac{p_{r(t)}}{q''_{r(t)}}$.
Using (\ref{logqn}, \ref{BS})  and (\ref{betas}), one gets
\begin{equation*}
y= y_0 +  \BS{\varphi}{r(t)}(\h{\alpha}) -  \BS{\varphi}{r(t)}(\h{\alpha}_0) = y_0 + \ln \frac{q''_{r(t)}}{q'_{r(t)}} = y_0 + \ln \frac{1+ \h{\alpha}^- \frac{p_{r(t)}}{q_{r(t)}}}{1+ \h{\alpha}_0^-  \frac{p_{r(t)}}{q_{r(t)}}} .
\end{equation*}
 As $t$, and hence $r(t)$, tend to infinity, $ \frac{p_{r(t)}}{q_{r(t)}}$ converge to $\alpha^+_0$ and we get (\ref{locunst}).

The global unstable and stable leaves can be obtained as
\begin{equation*}
\Gamma^{(u)}(\h{\alpha}_0, y_0)  = \bigcup_t \Phi_{t} \Gamma^{(u)}_{loc}(\h{\alpha}_{-t}, y_{-t}); \qquad \Gamma^{(s)}(\h{\alpha}_0, y_0)  = \bigcup_t \Phi_{-t} \Gamma^{(s)}_{loc}(\h{\alpha}_t, y_t).
\end{equation*}

To prove mixing, it is enough to show that the stable and unstable foliations form a non-integrable pair. From their non-integrability, it follows from the general theory  (see \cite{Si:top}) that the Pinsker partition is trivial and hence that $\{\Phi_t\}_{t\in \mathbb{R}}$ is a $K$-flow and, in particular,  is mixing.

Consider a sufficiently small neighborhood $\mathscr{U}(\h{\alpha_0}, y_0)\subset D_{\Phi}$ of $(\h{\alpha_0}, y_0)$. It is enough to show that, for a positive measure set of $(\h{\alpha}, y)\in \mathscr{U}(\h{\alpha_0}, y_0)$, $(\h{\alpha}, y)$ can be connected to $(\h{\alpha}_0, y_0)$ through a segments of local stable and unstable leaves, in particular if there exist  $(\h{\alpha}_i, y_i)\in \mathscr{U}(\h{\alpha_0}, y_0)$, $i=1,2$, such that $(\h{\alpha}_1, y_1) \in \Gamma^{(s)}(\h{\alpha_0}, y_0 )$, $(\h{\alpha}_2, y_2) \in \Gamma^{(u)}(\h{\alpha_1}, y_1 )$ and   $(\h{\alpha}, y) \in \Gamma^{(s)}(\h{\alpha_2}, y_2 )$.

Using explicitly the equations (\ref{locstab},\ref{locunst}), one can check that these points exist as soon as we can find $y_1$ and $\h{\alpha}_1^-$ such that $((\h{\alpha}_1^-, \h{\alpha}_0^+), y_1)\in \mathscr{U}(\h{\alpha_0}, y_0)$ and
\begin{equation}\label{findalpha1}
y_1 = y_0 + \ln \frac{1+\h{\alpha}_1^- \h{\alpha}_0+}{1+\h{\alpha}_0^- \h{\alpha}_0+}, \qquad y = y_1 + \ln \frac{1+\h{\alpha}_1^- \h{\alpha}+}{1+\h{\alpha}^- \h{\alpha}+},
\end{equation}
since in this case we can take $({\h{\alpha}}_1, y_1)  = ((\h{\alpha}_1^-, \h{\alpha}_0^+), y_1 )$ and $({\h{\alpha}}_2, y_2)  = ((\h{\alpha}_1^-, \h{\alpha}^+), y_1 )$. Equations (\ref{findalpha1}) can be solved if
\begin{equation}\label{hypsurf}
\frac{\h{\alpha}^+}{1+\h{\alpha}^- \h{\alpha}^+ } e^{y} \neq \frac{\h{\alpha}_0^+}{1+\h{\alpha}_0^- \h{\alpha}_0^+ } e^{y_0}.
\end{equation}
The points $(\h{\alpha}, y)$ for which there is the equality in (\ref{hypsurf}) lie on a surface in $D_{\Phi}$ and hence have measure zero.
This concludes the proof of the non integrability.
\end{proofof}

\section{Concluding Remark.}
Let $T$ be an ergodic automorphism of the measure space $(M,\mathscr{M}, \mu)$ and $f\in L^1(M,\mathscr{M}, \mu)$, $\int f\mathrm{d} x >0$. The following problem is a generalization of a classical renewal problem in probability theory. Take $R>0$ and consider the first $n_R$ such that
\begin{equation*} f(x)+ f(Tx) + \dots + f(T^{n_R} x)>R.\end{equation*}
 What will be the limiting distribution of  $f(x)+ f(Tx) + \dots + f(T^{n_R} x)- R$ as $R$ tends to infinity?
The answer can be given in terms of a special flow which is similar to the one considered above. Interesting aspects of this problem appear when $\int |f| \mathrm{d} x = \infty$.

Results concerning the limiting distribution when considering the Gauss map and the sum of the entries of the continued fraction expansion can be found in \cite{KS:dis}.

\subsection*{Acnkowledgments.}
The first author thanks NSF Grant DMS $0600996$ for the financial support.