text:
\section{Introduction}

A surface $L$ is a \emph{standard surface} if $L$ admits a complete hyperbolic metric which contains no hyperbolic  halfplanes and in which all the boundary components are geodesic~\cite[Definition~2]{cc:epstein}. A surface is of \emph{finite type} if it is of finite genus and has at most finitely many punctures and circle boundary components.

Throughout this paper $L$ is a fixed, standard hyperbolic surface of finite type and $f:L\to L$ is a fixed homeomorphism. Thus,  $L$ is assumed to have a complete hyperbolic structure, contains no hyperbolic halfplanes,  has finite genus, and at most finitely many cusps and geodesic circle boundary components.   We allow the surface $L$ to be nonorientable and the homeomorphism $f$ to be orientation reversing.

Let $\Gamma'$ be the collection of simple closed  geodesics $\gamma$  in $L\sm\bd L$ such that $f_{*}^{n}(\gamma) = \gamma$ for some $n>0$ (see Definition~\ref{fstar} for the definition of $f_{*}$). Let $\Gamma$ consist of those elements of $\Gamma'$ that are isolated in the sense that they are disjoint from all other elemants of $\Gamma'$. By the definition of $\Gamma$ and since $L$ has finite genus, $\Gamma$ consists of finitely many disjoint geodesics.  Since $f_{*}(\Gamma') = \Gamma'$, it follows that $f_{*}(\Gamma) = \Gamma$.

We prove,

\begin{theorem}\label{NTthm}

If $L$ be a connected, standard surface  of finite type and $f:L\to L$ a homeomorphism, then there exists a homeomorphism $h:L\to L$ isotopic to $f$ such that either,

\begin{enumerate}

\item  $h^{n}$ is isotopic to the identity for some integer $n>0$\upn{;}\label{NT1}

\item $h$ permutes a pair of transverse geodesic lamination $\Lambda_{\pm}$ which itersect every closed two sided  geodesic in $L\sm\bd L$ and are minimal in the sense that every leaf of each lamination is dense in the lamination\upn{;}\label{NT2}

\item If $S$ is the internal completion \emph{(Definition~\ref{intcompl})} of a component $U$ of $L\sm|\Gamma|$, then there is a least integer $n_{S}>0$ such that $h^{n_{S}}(S) = S$.  Further, $h^{n_{S}}|S:S\to S$ satisfies \emph{(\ref{NT1})} or \emph{(\ref{NT2})}.\label{part3}

\end{enumerate}

\end{theorem}

If $L$ is orientable,  then this is~\cite[Theorem~0.2]{ha:th}. We do not prove~\cite[Theorem~0.1]{ha:th} whose proof  uses different  ideas. We concentrate on constructing the laminations $\Lambda_{\pm}$. Our proof borrows  from Handel and Thurston~\cite{ha:th} and Casson and Bleiler~\cite{bca} but is substantially different from either proof.

\begin{rem}

If $S$ is the internal completion of a component $U$ of $L\sm|\Gamma|$, then $\bd S = S\sm U$  consists of geodesics. The inclusion map $U\to L$ extends in a natural way to an immersion $S\to L$  that sends these geodesics to geodesics in $\Gamma$. This immersion may send two geodesics in $\bd S$ to the same geodesic in $\Gamma$ or, if the image geodesic in $\Gamma$ is one sided, it will be the two-to-one image of a geodesic in $\bd S$.

\end{rem}

\begin{rem}[continued]

In Theorem~\ref{NTthm}~(\ref{part3}) we abuse notation and denote the natural extension of $h|U$ to $S$ again by $h$.

\end{rem}

\begin{defn}\label{supp}

If $\mathcal{G}$ is a set of  geodesics in $L$, we let $|\mathcal{G}|$ denote the \emph{support} of $\mathcal{G}$, that is the union of the elements of $\mathcal{G}$.

\end{defn}

\begin{defn}

If $U$ is   a component of $L\sm|\Gamma|$ which contains a simple closed geodesic $\sigma$ such that the geodesics $f_{*}^{n}(\sigma)$, $n\in\mathbb{Z}$, are distinct, then $U$ is called a \emph{pseuso-Anosov component}. Otherwise $U$ is called a \emph{periodic component}.

\end{defn}

Here $f_{*}$ is the geodesic tightening map (Definition~\ref{fstar}).

\begin{rem}

Since $f_{*}$ permutes the elements of $\Gamma$  we can extend the domain of definition of $f_{*}$ to  the components of $L\sm|\Gamma|$ so that $f_{*}$ permute the components of $L\sm|\Gamma|$.

\end{rem}

\begin{conv}

If $\gamma$ a geodesic in $\Delta$ and $\{\gamma_{n}\}_{n\ge0}$ a sequence of geodesics in $\Delta$ we  use the notation $\gamma_{n}\to\gamma$ as $n\to \infty$ for uniform convergence in the Euclidean metric on $\Delta$.

\end{conv}

\begin{defn}\label{defnLL}

Suppose $U$ is a pseudo-Anosov component  of $L\sm|\Gamma|$ and $\sigma\ss U\sm\bd L$ is a simple closed  geodesic. Define $\LL^{\sigma}_{+}$ (respectively $\LL^{\sigma}_{-}$) to be the set of simple geodesics $\gamma\ss U$ such that for every integer $p>0$,  there exists a subsequence $\{\sigma_{n_{k}p}\}_{k\ge 0}$ of $\{\sigma_{n}\}_{n\ge 0}$ (respectively $\{\sigma_{n}\}_{n\le 0}$) such that for any lift $\wt\gamma$ of $\gamma$ there exists a lift $\{\wt\sigma_{n_{k}p}\}_{k\ge 0}$ of $\{\sigma_{n_{k}p}\}_{k\ge 0}$ such that $\wt \sigma_{n_{k}p}\to\wt\gamma$ as $k\to\infty$.

\end{defn}

\begin{rem}

The full strength of Definition~\ref{defnLL} is needed in the proof of Lemma~\ref{perleaf}.

\end{rem}

In Section~\ref{constrLambdapm} we prove,

\begin{theorem}\label{proplams}

If $U$ is a pseudo-Anosov component  of $L\sm|\Gamma|$ and $\sigma\ss U\sm\bd L$ is a simple closed  geodesic, then $\sigma_{n} = f_{*}^{n}(\sigma)$, $n\in\mathbb{Z}$, are distinct, $\emptyset\ne|\LL^{\sigma}_{\pm}|\ss U$,  and the geodesics in $\LL^{\sigma}_{\pm}$ are  homeomorphic to the reals.   Further,

\begin{enumerate}

\item There exists a unique minimal geodesic laminations $\Lambda_{\pm}\ss\LL^{\sigma}_{\pm}$\upn{;}

\item $\Lambda_{+}$ and $\Lambda_{-}$ are transverse\upn{;}

\item The components of $U\sm(|\Lambda_{+}|\cup|\Lambda_{-}|)$ consist  of open rectangles and finitely many open disks and/or open or half open annuli\upn{;}

\item The laminations $\Lambda_{\pm}$ are independent of the choice of $\sigma$\upn{;}

\item $\LL^{\sigma}_{\pm}\sm\Lambda_{\pm}$ are each finite sets\upn{;}

\end{enumerate}

\end{theorem}

\section{Definitions and preliminary results}\label{defres}

We take   the open unit disk $\Delta$ with the Poincar\'e metric as hyperbolic plane. The closed unit disk $\mathbb{D}^{2} = \Delta\cup\Si$, where the  unit circle $\Si$ is called the \emph{circle at infinity}.

Throughout, $L$ is a connected, standard surface  of finite type. We fix a lift $\wt L$ to $\Delta$ and let $\Se = \bd\wt L\ss\mathbb{D}^{2}$. We denote a lift of $f:L\to L$ by $\wt f:\wt L\to\wt L$.  We denote by $\wh L$ the closure of $\wt L$ in $\mathbb{D}^{2}$ and refer to $E = \wh L\cap\Si$ as the \emph{ideal boundary} of $L$. By~\cite[Theorem~2]{cc:epstein}, the map   $\wt f:\wt L\to\wt L$ has a natural extension $\wh f:\wh L\to\wh L$.

\subsection{Surfaces of finite type}\label{fintype}

The following propositions are well known result in hyperbolic geometry.

\begin{prop}\label{areahypsur}

If $L$ is a standard hyperbolic of finite type with $c\ge0$ cusps, $m\ge0$ cross caps, $b\ge 0$ circle boundary components, and $g\ge 0$ handles, then $${\rm area}\,(L) = 2\pi(c+m+b+2g-2).$$

\end{prop}

\begin{rem}

By~\cite[Theorem~8]{cc:epstein}, if $L$ is a standard hyperbolic surface,  then $$c+m+b+2g\ge3.$$

\end{rem}

\begin{prop}

Each cusp of $L$ has a neighborhood such that no simple closed geodesic in $L$ meets that neighborhood.\label{cuspnhb}

\end{prop}

\subsection{Pseudo-geodesics and geodesic tightening}

\begin{defn}[pseudo-geodesic]\label{pseudogeodesic}
A curve  $\gamma\ss L$ is a  \emph{pseudo-geodesic} if either some (hence every) lift $\wt\gamma$ has two distinct, well defined endpoints on $\Si$  or $\gamma$ is a properly embedded, boundary incompressible  compact arc.
\end{defn}

Remark that geodesics and  essential embedded circles in $L$ that do not bound cusps   are pseudo-geodesics and that if $f$ is continuous and $\gamma$ is a geodesic, then by~\cite[Theorem~2]{cc:epstein}  $f(\gamma)$ is a pseudo-grodesic.

\begin{defn}

If $\gamma$ is a pseudo-geodesic, then the \emph{geodesic tightening} of $\gamma$ is the unique geodesic whose lifts have the same endpoints on $\Se$ as the lifts of $\gamma$.

\end{defn}

\begin{defn}\label{fstar}

If $\gamma$ is a geodesic, we will demote the geodesic tightening of $f(\gamma)$ by $f_{*}(\gamma)$. If $\mathcal{G}$ is a set of geodesics, we define
$$f_{*}(\mathcal{G}) = \{f_{*}(\gamma)\ |\ \gamma\in\mathcal{G}\}.$$

\end{defn}

The next lemma follow immediately from~\cite[Theorem~2]{cc:epstein}.

\begin{lemma}\label{f*seq}

If $\gamma_{k}$, $k\ge0$, and $\gamma$ are geodesics with lifts $\wt\gamma_{k}$, $k\ge0$, and $\wt\gamma$ such that $\wt\gamma_{k}\to\wt\gamma$ as $k\to\infty$ and $\wt f_{*}$ is a lift of $f_{*}$, then $\wt f_{*}(\gamma_{k})\to\wt f_{*}(\wt\gamma)$ as $k\to\infty$.

\end{lemma}

\subsection{Laminations}

\begin{defn}

A \emph{lamination} $\Lambda$ is a set of simple geodesics in $L$ whose support $|\Lambda|$  is closed in $L$.  A geodesic in $\Lambda$ is called a \emph{leaf} of $\Lambda$.

\end{defn}

\begin{defn}

A nonempty lamination which properly contains no other lamination is said to be \emph{minimal}.

\end{defn}

\begin{defn}[semi-isolated]\label{defnsemiis}
If a leaf $\lambda\in\Lambda$ is approached by points of  $|\Lambda|$ on at most one side, we say $\lambda$ is \emph{semi-isolated}.
\end{defn}

\begin{rem}

Note that our definition of semi-isolated includes all isolated leaves.

\end{rem}

\subsection{Internal completion}

Let $\Lambda$ be a lamination in $L$. A component $V$  of $L\sm|\Lambda|$ has two different hyperbolic metrics, the first defined by geodesics of $L$ and the second defined by segments of  geodesics of $L$ contained in $V$.

\begin{defn}\label{intcompl}

Define the \emph{internal completion} of a component $V$ of $L\sm|\Lambda|$ to be the metric completion of $V$ in the metric defined  by segments of  geodesics of $L$ contained in $V$.

\end{defn}

\begin{defn}

If $V$ is a component of $L\sm|\Lambda|$, define a \emph{border component} of $v$  to be a boundary component of the internal completion of $V$. Let $\delta V$ denote the set of border compoents of $V$.    The set $|\delta V|$ is called the \emph{border} of $V$.

\end{defn}

\begin{lemma}

The point $x\in|\delta V|$ if and only if there exists a segment $[x,y]$ of a geodesic of $L$ such that $(x,y]\ss V$ and $x\notin V$.

\end{lemma}

There is a natural map from the internal completion of a component $V$ of $L\sm|\Lambda|$ into $L$ taking $\delta V$ to a set of semi-isolated leaves of $\Lambda$. An isolated leaf of $\lambda$ might be the image of two border components of $V$ under this map but  a semi-isolated leaf of $\Lambda$ which is not isolated is the image of at most one.

\begin{rem}

In abuse of notation we will often identify $\delta V$ with its image under this map and consider $\delta V$ to be a subset of $\Lambda$.

\end{rem}

\begin{rem}

For more information on border and internal completion see~\cite[Section~5.1]{cc:hm}.

\end{rem}

\section{Simple closed geodesics in $L$}\label{simplegeo}

Let $\sigma$ be a simple closed geodesic in $L$. Note that since $\sigma$ is a geodesic, $\sigma$ is essential and can not bound a cusp.  Throughout this paper, we use the notation,

\begin{nota}

 $\sigma_{n} = f_{*}^{n}(\sigma)$, $n\in\mathbb{Z}$.

 \end{nota}

\begin{lemma}

If there exists $n\in\mathbb{Z}$ and $k>0$ such that $\sigma_{n} = \sigma_{n+k}$,
then $\sigma_{m} = \sigma_{m+k}$, all $m\in\mathbb{Z}$.

\end{lemma}

\begin{proof}

$\sigma_{m} = f_{*}^{m-n}\circ f_{*}^{n}(\sigma) = f_{*}^{m-n}\circ f_{*}^{n+k}(\sigma) = \sigma_{m+k}.$
\end{proof}

Thus, one of the following two possibilities holds,

\begin{enumerate}

\item $\sigma_{k} = \sigma$ some least integer $k>0$;

\item $\sigma_{n}$, $n\in\mathbb{Z}$, are distinct.

\end{enumerate}

The following lemma is clear.

\begin{lemma}

If $\gamma$ is a simple closed geodesic and $\tau$ is any simple geodesic, then $\tau$ spirals on $\gamma$ if and only if there are lifts $\wt\gamma$ of $\gamma$ and $\wt\tau$ of $\tau$ sharing an endpoint on $\Si$.

\end{lemma}

\begin{prop}\label{spirals}

If $\gamma$ is a simple closed geodesic, then there exists $\epsilon>0$ such that if  $\tau$ is a simple geodesic $\tau$ containing a point at a distance less than $\epsilon$ from $\sigma$, then either,

\begin{enumerate}

\item $\tau$ spirals on $\gamma$ or\upn{;}

\item $\gamma$ and $\tau$ intersect transversely.

\end{enumerate}

\end{prop}

\begin{proof}

Compare~ the proof of \cite[Lemma~4.6]{bca}.   Choose a lift $\wt\gamma$ of $\gamma$. Let   $T_{\wt\gamma}$ be an orientation preserving deck transformation with $\wt\gamma$ as axis and $d$ the hyperbolic distance a point of $\wt\gamma$ is moved by $T_{\wt\gamma}$. Choose $\epsilon>0$ such that if $\sigma$ is any geodesic in $\Delta$ disjoint from $\wt\gamma$, not sharing an endpoint on $\Si$ with $\wt\gamma$, and     containing a point at a distance less than $\epsilon$ from $\wt\gamma$, then the perpendicular projection (see~\cite[Figure~4.5]{bca})  of $\sigma$ on $\wt\gamma$ has length greater than $d$.

Suppose that $\tau$ is a simple geodesic that does not spiral on $\gamma$, does not intersect $\gamma$, and does contain a point at a distance  less than $\epsilon$ from $\gamma$. Then $\tau$ has a lift $\wt\tau$ containing a point at a distance less than $\epsilon$ from $\wt\gamma$, does not intersect $\wt\gamma$, and does not share an endpoint on $\Si$ with $\wt\gamma$. It follows that the perpendicular projection of $\wt\tau$ on $\wt\gamma$ has length greater than $d$. Thus, $T_{\wt\gamma}(\wt\tau)$ meets $\wt\tau$ contradicting the fact that $\tau$ is simple. It follows that either $\tau$ intersects $\gamma$ or that $\tau$ spirals on $\gamma$.

The proof of the converse is immediate.
\end{proof}

For the rest of Section~\ref{simplegeo} we make the following assumption.

\begin{*assump}\label{assumeone}

Assume that,

\begin{enumerate}

\item $\gamma$ and $\sigma$ are simple closed geodesics\upn{;}

\item There exist a subsequence $\{\sigma_{n_{k}}\}_{k\ge0}$ of $\{\sigma_{n}\}_{n\ge0}$ and lifts $\wt\gamma$ of $\gamma$ and $\wt\sigma_{n_{k}}$ of $\sigma_{n_{k}}$ such that $\wt\sigma_{n_{k}}\to\wt\gamma$ as $k\to\infty$.

\end{enumerate}

\end{*assump}

By Proposition~\ref{spirals}, we can assume that all the $\sigma_{n_{k}}$ intersect $\gamma$ transversely,

\begin{cor}\label{spiral}

If $\gamma$ and $\sigma$ satisfy  the above \emph{Assumption}, then there exists a subsequence $\{\sigma_{n_{\ell}}\}_{\ell\ge0}$ of  $\{\sigma_{n_{k}}\}_{k\ge0}$ and a geodesic $\tau$ such that each $\sigma_{n_{\ell}}$ has lift $\wt\sigma'_{n_{\ell}}$ with $\wt\sigma'_{n_{\ell}}\to\wt\tau$, a lift of $\tau$, and $\tau$ spirals on $\gamma$.

\end{cor}

\begin{proof}

Denote by $T_{\wt\gamma}$ an orientation preserving deck transformation with axis $\wt\gamma$ and let $[a,b]\ss\Si$ be such that $T_{\wt\gamma}(a) = b$. Each $\sigma_{n_{k}}$ has a lift meeting $\wt\gamma$ with an endpoint in $[a,b)$ so there exists a subsequence $\{\sigma_{n_{\ell}}\}_{\ell\ge0}$ of $\{\sigma_{n_{k}}\}_{k\ge0}$ and a geodesic $\tau$ such that each $\sigma_{n_{\ell}}$ has lift $\wt\sigma'_{n_{\ell}}$ meeting $\wt\gamma$ with an endpoint in $[a,b)$ and $\wt\sigma'_{n_{\ell}}\to\wt\tau$, a lift of $\tau$ with endpoint in $[a,b]$.

The geodesics $\tau$ and $\gamma$ can not intersect since each $\sigma_{n_{\ell}}$ is simple and there exist sequences of lifts of the $\sigma_{n_{\ell}}$ approaching both $\wt\tau$ and $\wt\gamma$.  Since each $\wt\sigma'_{n_{\ell}}$ meets $\wt\gamma$ and $\wt\sigma'_{n_{\ell}}\to\wt\tau$, it follows that $\wt\gamma$ and $\wt\tau$ share an endpoint on $\Si$. Thus $\tau$ spirals on $\gamma$.
\end{proof}

\begin{nota}

The notation $\#(A)$ denotes the number of points in the finite set $A$.

\end{nota}

\begin{cor}

If $\gamma$ and $\sigma$ satisfy  the above \emph{Assumption}, then  for each integer $r>0$, $f^{r}_{*}(\gamma)$ and $\gamma$ are either disjoint or coincide.

\end{cor}

\begin{proof}

Let $N_{r} = \#(\sigma\cap  \sigma_{r}) = \#(\sigma_{n_{\ell}}\cap \sigma_{r+n_{\ell}})$ and $\tau$ be the geodesic constructed in Corollary~\ref{spiral}.  If $\gamma$ and $f_{*}^{r}(\gamma)$ intersect transversly, then $\tau$ and $f_{*}^{r}(\tau)$ have infinitely many points of transverse intersection.  Choose disjoint neighborhoods $V_{j}$, $1\le j\le N_{r}+1$, of $N_{r}+1$ points of transverse intersetion of $\tau$ and $f_{*}^{r}(\tau)$. By taking $\ell$ sufficiently large one can find a point of $\sigma_{n_{\ell}}\cap \sigma_{r+n_{\ell}}$ in each $V_{j}$. Since there are $N_{r}$ points in $\sigma_{n_{\ell}}\cap \sigma_{r+n_{\ell}}$ and $N_{r}+1$ of the $V_{j}$ this is a contradiction and the lemma follows.
\end{proof}

The next corollary follows since $L$ has finite genus.

\begin{cor}\label{inGamma}

If $\gamma$ and $\sigma$ satisfy  the above \emph{Assumption},  then for some least integer $r>0$, $f^{r}_{*}(\gamma) = \gamma$, $f_{*}^{r}$ fixes  the orientation of $\gamma$, and $f^{r}$ is orientation preserving.

\end{cor}

Let $c,d$ be the endpoints of $\wt\gamma$ and $\wt f_{*}^{r}$ be a lift of $f_{*}^{r}$ fixing $\wt\gamma$ and thus both $c$ and $d$.

\begin{cor}\label{transto}

If $\gamma$ and $\sigma$ satisfy  the above \emph{Assumption} and  $\alpha$ is any simple closed geodesic transverse to $\gamma$, then $f^{n}_{*}(\alpha)$, are distinct, $n\in\mathbb{Z}$.

\end{cor}

\begin{proof}

Suppose $f^{n}_{*}(\alpha) = \alpha$ for some $n>0$   and       choose the integer $r$ and lifts $\wt\gamma$ and $\wt f_{*}^{r}$ as above . By choosing $r$ larger, we can further assume that $\wt f^{r}_{*}(\wt\alpha) = \wt\alpha$ for every lift $\wt\alpha$ of $\alpha$ meeting $\wt\gamma$. By Proposition~\ref{spirals}, we can assume that $\wt\sigma_{n_{k}}\cap\wt\gamma\ne0$ all $k\ge 1$. Let $T_{\wt\gamma}$ be aa orientation preserving deck transformation with axis $\wt\gamma$. Choose $x\in\wt\gamma$ and let $y = T_{\wt\gamma}(x)$. Let $\mathcal{G}$ be the finite set of lifts of the geodesics $\sigma_{0},\ldots,\sigma_{r-1}$ meeting the interval $[x,y]\ss\wt\gamma$. Since $\mathcal{G}$ is finite, there exists lifts $\wt\alpha$ and $\wt\alpha'$ of $\alpha$ meeting $\wt\gamma$ such that all the geodesic in $\mathcal{G}$ lie between $\wt\alpha$ and $\wt\alpha'$ in $\Delta$.

Consider $\sigma_{n}$ where $n = i + jr$ for integers $0\le i\le r-1$ and $j\ge 0$. Then, $\sigma_{n} = f_{*}^{jr}(\sigma_{i})$. If $\beta$ is a lift of $\sigma_{n}$ that meets $\wt\gamma$, let $\rho =  \wt  f_{*}^{-jr}(\beta)$. Then $\rho$ is a lift of $\sigma_{i}$  that meets $\wt\gamma$. Thus, there exists an integer $s\in\mathbb{Z}$, such that $\rho = T_{\wt\gamma}^{s}(\zeta)$ with $\zeta\in\mathcal{G}$ a lift of $\sigma_{i}$ meeting the interval $[x,y]\ss\wt\gamma$. Since $\zeta$ is between  the lifts $\wt\alpha$ and $\wt\alpha'$   of $\alpha$, it follows that $\rho$ is between   the lifts $T_{\wt\gamma}^{s}(\wt\alpha)$ and $T_{\wt\gamma}^{s}(\wt\alpha')$  of $\alpha$. Thus, $\beta = \wt  f_{*}^{jr}(\rho)$ is between $\wt  f_{*}^{jr}\circ T_{\wt\gamma}^{s}(\wt\alpha) = T_{\wt\gamma}^{s}(\wt\alpha)$ and $\wt  f_{*}^{jr}\circ T_{\wt\gamma}^{s}(\wt\alpha') = T_{\wt\gamma}^{s}(\wt\alpha')$. That is,  any lift of $\sigma_{n}$, $n\in\mathbb{Z}$, meeting   $\wt\gamma$ lies  between $T_{\wt\gamma}^{s}(\wt\alpha)$ and $T_{\wt\gamma}^{s}(\wt\alpha')$, for some $s\in\mathbb{Z}$.

Therefore, given $k\ge 0$, there exsts an integer $s_{k}\in\mathbb{Z}$ such that $\wt\sigma_{n_{k}}$ lies between $T_{\wt\gamma}^{s_{k}}(\wt\alpha)$ and $T_{\wt\gamma}^{s_{k}}(\wt\alpha')$, some $s_{k}\in\mathbb{Z}$. Thus, the sequence $\{\wt\sigma_{n_{k}}\}_{k\ge0}$ does not converge to $\wt\gamma$. This contradicts our Assumption and the  corollary follows.
\end{proof}

\section{The  laminations $\Lambda^{\sigma}_{\pm}$}\label{constrLambdapm}

\begin{*assump}\label{assumetwo}

Throughout Section~\ref{constrLambdapm},   $U$ is a fixed component of $L\sm|\Gamma|$ and $\sigma\ss U$ is a fixed geodesic such that the geodesics $\sigma_{n} = f_{*}^{n}(\sigma)$, $n\in\mathbb{Z}$, are distinct.

\end{*assump}

\subsection{$\LL^{\sigma}_{+}$ and the positive  lamination $\Lambda^{\sigma}_{+}$}\label{poslam}

In this and the next section we explicitly prove results for $\LL^{\sigma}_{+}$ and the positive lamination $\Lambda^{\sigma}_{+}$. The same results hold for $\LL^{\sigma}_{-}$ and the negative lamination $\Lambda^{\sigma}_{-}$ with the obvious modifications.

\begin{nota}

Let $n_{U}$ denote the least positive integer such that $f_{*}^{n_{U}}(U) = U$.

\end{nota}

\begin{lemma}

$\LL^{\sigma}_{+}\ne\emptyset$.

\end{lemma}

\begin{proof}
By Proposition~\ref{cuspnhb} there exists a compact subsurface $L'\ss L$   such that every simple closed geodesic in $L$ is contained in $L'$. Define,
$$X^{\sigma}_{+}(p) = \bigcap_{n=1}^{\infty}\ol{\bigcup_{k=n}^{\infty}f_{*}^{p!n_{U}k}(\sigma)}\ss L'\cap U.$$
Note that,
$$X^{\sigma}_{+}(1)\supset X^{\sigma}_{+}(2)\supset\cdots\supset X^{\sigma}_{+}(p)\supset\cdots.$$
Define,
$$X^{\sigma}_{+} = \bigcap_{p=1}^{\infty}X^{\sigma}_{+}(p)\ss L'\cap U.$$
The set $X^{\sigma}_{+}\ne\emptyset$ by the compactness of $L'$. We now show that $X^{\sigma}_{+}\ss|\LL^{\sigma}_{+}|$.

Suppose $x\in X^{\sigma}_{+}$ and $p>0$ is an integer. We will show that there exists a simple geodesic $\beta\in\LL^{\sigma}_{+}$ containing $x$.

Since $x\in X^{\sigma}_{+}$, there exists a subsequence $\{\sigma_{n_{k}p}\}_{k\ge 0}$ of $\{\sigma_{p!n_{U}n}\}_{n\ge 0}$ and points $x_{k}\in\sigma_{n_{k}p}$ such that $x_{k}\to x$. Let $v_{k}$ be the unit tangent vector to $\sigma_{n_{k}p}$ at $x_{k}$. By again passing to a subsequence if neccesary we can assume that $v_{k}\to v$, a unit tangent vector to $L$ at $x$. Let $\beta$ be the unique geodesic through $x$ with tangent vector $v$ and $\wt\beta$ any lift of $\beta$. Let $\wt x$ be the lift of $x$ in $\wt\beta$ and choose lifts $\wt x_{k}$ of $x_{k}$ with $\wt x_{k} \to \wt x$. Choose lifts $\wt\sigma_{n_{k}p}$ of $\sigma_{n_{k}p}$ containing $\wt x_{k}$, $k\ge 0$.   Then $\wt \sigma_{n_{k}p}\to \wt\beta$ so $\beta\in\LL^{\sigma}_{+}$.

If $\beta$ is not simple, then $\beta$ has lifts $\wt\beta$ and $\wt\beta'$ such that $\wt\beta\cap\wt\beta'\ne\emptyset$. Then for $k$ sufficiently large, $\sigma_{n_{k}p}$ has lifts $\wt\sigma_{n_{k}p}$ and $\wt\sigma_{n_{k}p}'$ such that $\wt\sigma_{n_{k}p}\cap\wt\sigma_{n_{k}p}'\ne\emptyset$ so $\sigma_{n_{k}p}$ is not simple. This contradiction implies that $\beta$ is simple.

Thus, $x\in\beta\in\LL^{\sigma}_{+}$ so $x\in|\LL^{\sigma}_{+}|$.
\end{proof}

\begin{prop}\label{homeoR}

Every geodesic in $\LL^{\sigma}_{+}$ is homeomorphis to the reals.

\end{prop}

\begin{proof}

If  $\gamma\in\LL^{\sigma}_{+}$, then  there exists a lift $\wt\gamma$ of $\gamma$ and  a sequence $\{\sigma_{n_{k}}\}_{k\ge0}$ with lift $\{\wt\sigma_{n_{k}}\}_{k\ge0}$  such that $\wt \sigma_{n_{k}}\to\wt\gamma$ as $k\to\infty$. If $\gamma$ is closed, by Proposition~\ref{spirals}, we can assume that the $\sigma_{n_{k}}$ meet $\gamma$ transversely. Corollary~\ref{inGamma} then implies that $\gamma\in\Gamma'$. By Corollary~\ref{transto} it follows that $\gamma\in\Gamma$. Since $\sigma\ss U$, it follows that $\sigma$ and each $\sigma_{n}$, $n\in\mathbb{Z}$, are disjoint from each geodesic in $\Gamma$ and thus from $\gamma$. This contradiction implies the proposition.
\end{proof}

Since $\delta U$ consists of closed geodesics we have,

\begin{cor}\label{ssU}

$|\LL^{\sigma}_{+}|\ss U$.

\end{cor}

Let $\beta\in\LL^{\sigma}_{+}$. Recall that the closure of a geodesic is the disjoint union of geodesics.  If $\Lambda_{\beta}$ denotes the lamination consisting of the geodesic in $\ol\beta$, then,

\begin{lemma}\label{convto}

 $\Lambda_{\beta}\ss\LL^{\sigma}_{+}$.

\end{lemma}

\begin{proof}

Suppose $\gamma\in\Lambda_{\beta}$, $\wt\gamma$ is a lift of $\gamma$,  and $p>0$ is an integer.  Since $\beta\in\LL^{\sigma}_{+}$, there exists a subsequence $\{\sigma_{n_{k}p}\}_{k\ge0}$ of $\{\sigma_{n}\}_{n\ge0}$ such that if $\wt\beta$ is a lift of $\beta$ there exists a lift $\{\wt\sigma_{n_{k}p}\}_{k\ge0}$  of $\{\sigma_{n_{k}p}\}_{k\ge0}$ such that $\wt\sigma_{n_{k}p}\to\wt\beta$ as $k\to\infty$. Fix the sequence $\{\sigma_{n_{k}p}\}_{k\ge0}$  but not the sequence $\{\wt\sigma_{n_{k}p}\}_{k\ge0}$  of lifts.

Since $\gamma\ss\ol\beta$, we can choose lifts $\wt\beta_{\ell}$ of $\beta$ such that each endpoint on $\Si$ of $\wt\beta_{\ell}$ is at a distance less than $1/\ell$ from the corresponding endpoint of $\wt\gamma$. Inductively, suppose integers $k_{1}<\cdots<k_{\ell-1}$ and lfts $\wt\sigma_{n_{k_{i}}p}$ of $\sigma_{n_{k_{i}}p}$
have been chosen so that the corresponding endpoints on $\Si$ of $\wt\beta_{i}$ and $\wt\sigma_{n_{k_{i}}p}$ are at a distance less than $1/i$, $1\le i\le\ell-1$. Since each $\wt\beta_{\ell}$ is a lift of $\beta$, there exists  an integer $k_{\ell}>k_{\ell}-1$ and lift $\wt\sigma_{n_{k_{\ell}}p}$ of $\sigma_{n_{k_{\ell}}p}$ such that   the corresponding endpoints on $\Si$ of $\wt\beta_{\ell}$ and $\wt\sigma_{n_{k_{\ell}}p}$ are at a distance less than $1/\ell$. Then $\wt\sigma_{n_{k_{\ell}}p}\to\wt\gamma$ as $\ell\to\infty$. Let $\wt\gamma'$ be any lift of $\gamma$. Then there exists a deck transformation $T$ so that $\wt\gamma' = T(\wt\gamma)$. Then $\{T(\wt\sigma_{n_{k_{\ell}}p})\}_{\ell\ge0}$ is a   lift of the sequence $\{\sigma_{n_{k_{\ell}}p}\}_{\ell\ge0}$ and $T(\wt\sigma_{n_{k_{\ell}}p})\to T(\wt\gamma) = \wt\gamma'$ as $\ell\to\infty$ so $\gamma\in\LL^{\sigma}_{+}$ as desired.
\end{proof}

By compactness of $L'$ we have,

\begin{cor}

There exist minimal laminations that are subsets of $\LL^{\sigma}_{+}$.

\end{cor}

\begin{defn}

Let  $\Lambda^{\sigma}_{+}$ be a fixed minimal lamination which is a subset of $\LL^{\sigma}_{+}$.

\end{defn}

We show in Proposition~\ref{lamuniq} that the choice of $\Lambda^{\sigma}_{+}$ is unique.

\begin{lemma}

The space  $|\Lambda^{\sigma}_{+}|$ has empty interior. The geodesics in $\Lambda^{\sigma}_{+}$  are the path components of $|\Lambda^{\sigma}_{+}|$. Further, $|\Lambda^{\sigma}_{+}| = \ol{\lambda}$ for any $\lambda\in\Lambda^{\sigma}_{+}$.

\end{lemma}

\begin{proof}

If $|\Lambda^{\sigma}_{+}|$ has interior, then $|\Lambda^{\sigma}_{+}|$ is a surface without boundary  because the closure of any boundary component would be a  proper sublamination of $\Lambda^{\sigma}_{+}$ contradicting the fact that $\Lambda^{\sigma}_{+}$ is minimal. Thus, $|\Lambda^{\sigma}_{+}| = L$ is a foliated surface with negative Euler characteristic which is a contradiction. Thus, $|\Lambda^{\sigma}_{+}|$ has empty interior. It follows that the path componets of $|\Lambda^{\sigma}_{+}|$ are the geodesics in $\Lambda^{\sigma}_{+}$. That $|\Lambda^{\sigma}_{+}| = \ol{\lambda}$ for any $\lambda\in\Lambda^{\sigma}_{+}$ follows since $\Lambda^{\sigma}_{+}$ is minimal.
\end{proof}

The next corollary follows since $\Lambda^{\sigma}_{+}$ is a minimal lamination containing no  closed geodesic.

\begin{cor}

 $\Lambda^{\sigma}_{+}$ is transversely a Cantor set.

\end{cor}

\begin{lemma}

Suppose $\wt \sigma_{n_{k}}\to\wt\lambda$ where $\ol\lambda = |\Lambda^{\sigma}_{+}|$.   Then $\wt \sigma_{n_{k}+r}\to\wt\lambda_{r} = \wt{f_{*}^{r}(\lambda)}$ and $\ol\lambda_{r} = |f_{*}^{r}(\Lambda^{\sigma}_{+})|$.

\end{lemma}

\begin{lemma}\label{disjlams}

$\Lambda^{\sigma}_{+}$ and $f_{*}^{r}(\Lambda^{\sigma}_{+})$ have no transverse points of intersection.  Since $\Lambda^{\sigma}_{+}$ is minimal it follows that $\Lambda^{\sigma}_{+}$ and $f_{*}^{r}(\Lambda^{\sigma}_{+})$ are either disjoint or coincide.

\end{lemma}

\begin{proof}

Let $N_{r} = \#(\sigma\cap  \sigma_{r}) = \#(\sigma_{n_{k}}\cap \sigma_{r+n_{k}})$. If $\Lambda^{\sigma}_{+}$ and $f_{*}^{r}(\Lambda^{\sigma}_{+})$ have one point of transverse intersection, they have infinitely many points of transverse intersection.  Choose disjoint neighborhoods $V_{j}$, $1\le j\le N_{r}+1$, of $N_{r}+1$ points of transverse intersetion of $\Lambda^{\sigma}_{+}$ and $f_{*}^{r}(\Lambda^{\sigma}_{+})$. By taking $k$ sufficiently large one can find a point of $\sigma_{n_{k}}\cap \sigma_{r+n_{k}}$ in each $V_{j}$. Since there are $N_{r}$ points in $\sigma_{n_{k}}\cap \sigma_{r+n_{k}}$ and $N_{r}+1$ of the $V_{j}$ this is a contradiction and the lemma follows.
\end{proof}

\subsection{Crown sets}\label{crsets}

If $V$ is a  component of $L\sm|\Lambda^{\sigma}_{+}|$, then a geodesic $\gamma^{+}_{0}\in\delta V$ determines a bi-infinite sequence $\{\gamma^{+}_{i}\}_{i\in\mathbb{Z}}\ss\delta V$ with lift $\{\wt\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$ such that the terminal end of $\wt\gamma^{+}_{i}$ and initial end of $\wt\gamma^{+}_{i+1}$  share the endpoint $a_{i}$ on $\Si$.

\begin{lemma}

There are finitely many components $V$ of $L\sm|\Lambda^{\sigma}_{+}|$  and for each such $V$, $\delta V$ is a finite set.

\end{lemma}

\begin{proof}

Let $V$ be a component of $L\sm|\Lambda^{\sigma}_{+}|$,  $X$ be the internal completion of $V$, and  $DX$ denote the double of $X$. Then,  by~\cite[Lemma 1]{cc:epstein},  $DX$ is a standard hyperbolic surface. If $\delta V$ is an infinite set, then $DX$ has infintely many cusps and thus infinite area contradicting the fact that $L$ is of finite type.  By Proposition~\ref{areahypsur} and the remark in Section~\ref{fintype}, ${\rm area}\,(DX)\ge2\pi$ so ${\rm area}\,(X)\ge\pi$. Since ${\rm area}\,(L)$ is finite, it follows that there are finitely many components of the complement of $|\Lambda^{\sigma}_{+}|$.
\end{proof}

Let $V$ be a component of $L\sm|\Lambda^{\sigma}_{+}|$ and $\{\gamma^{+}_{i}\}_{i\in\mathbb{Z}}\ss\delta V$  with lift $\{\wt\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$  be as above. Let $\wt V$ be the lift of $V$ with $\{\wt\gamma^{+}_{i}\}_{i\in\mathbb{Z}}\ss\delta\wt V$.  Then the covering projection $\wt V\to V$ maps  the closure of the convex hull of $\{\wt\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$ onto a set $C_{+}\ss V$.

\begin{defn}

The set $C_{+}$    is called the \emph{crown set associated} to $\{\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$.

\end{defn}

\begin{lemma}

A crown set is either homeomorphic to,

\begin{enumerate}

\item A disk with $p$ points removed from the  boundary\upn{;}

\item A punctured disk with $p$ points removed from the  boundary\upn{;}

\item An annulus with $p$ points removed from one boundary\upn{;}

\item A M\"obius strip with $p$ points removed from its boundary.

\end{enumerate}

\end{lemma}

\begin{proof}

If $\{\wt\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$ is a finite set as in Figure~\ref{CSleft}, then $C_{+}$ is a disk with $p$ points removed from the  boundary. Otherwise, $\{\wt\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$ is an infinite set. The points $a_{i}$, $i\in\mathbb{Z}$, limit on two points in $\Si$. If these two points coincide as in Figure~\ref{CScusp}, then $C_{+}$ is a punctured disk with $p$ points removed from the  boundary. If these two points are distinct as in Figure~\ref{CSright}, then $C_{+}$ is either an annulus with $p$ points removed from one boundary or a M\"obius strip with $p$ points removed from its boundary depending on whether the geodesic $\wt\rho\ss\Delta$ with endpoints the two limit points projects under the covering projection to a two sided or one sided simple closed curve  $\rho\ss L$.
\end{proof}

\begin{figure}[t]
\begin{center}
\begin{picture}(200,140)(-20,-140)
\rotatebox{270}{\includegraphics[width=140pt]{CrownSetLeft}}

\put (-77,-55){\small$\wt C_{+}$}
\put (-102,-85){\small$\tilde{\gamma}^{+}_{0}$}
\put (-47,-85){\small$\tilde{\gamma}^{+}_{2}$}
\put (-74,-105){\small$\tilde{\gamma}^{+}_{1}$}
\put (-140,-96){\small$b_{0}$}
\put (-3,-90){\small$b_{2}$}
\put (-70,-145){\small$b_{1}$}
\put (-32,-128){\small$a_{1}$}
\put (-114,-129){\small$a_{0}$}

\end{picture}
\caption{Lift of simply connected crown set $C_{+}$}\label{CSleft}
\end{center}
\end{figure}

\begin{defn}

If the crown set $C_{+}$ is an annulus with $p$ points removed from one boundary, then $C_{+}$ is called a \emph{crown}~\cite[Figure~4.2]{bca} and the simple closed curve boundary component $\rho$ of $C_{+}$ is called the \emph{rim} of the crown $C_{+}$.

\end{defn}

\begin{rem}

In the case that $C_{+}$ is a M\"obius strip with $p$ points removed from its boundary, the geodesic  $\wt\rho\ss\Delta$ projects under the covering projection to the center geodesic of the M\"obius strip.

\end{rem}

\begin{figure}[h]
\begin{center}
\begin{picture}(180,40)(60,-40)
\rotatebox{270}{\includegraphics[width=50pt]{CrownSetCusp}}

\put (-125,-3){\small$\tilde{C_{+}}$}
\put (-159,2){\small$\tilde{\gamma}^{+}_{1}$}
\put (-228,2){\small$\tilde{\gamma}^{+}_{0}$}
\put (-89,2){\small$\tilde{\gamma}^{+}_{2}$}
\put (-228,-49){\small$b_{0}$}
\put (-89,-49){\small$b_{2}$}
\put (-157,-49){\small$b_{1}$}
\put (-192,-49){\small$a_{0}$}
\put (-124,-49){\small$a_{1}$}
\put (-30,-30){$\dots$}
\put (-290,-30){$\dots$}

\end{picture}
\caption{Lift of crown set $C_{+}$ containing a cusp}\label{CScusp}
\end{center}
\end{figure}

Let $R^{\sigma}$ be the set consisting of all rims of  crowns of $\Lambda^{\sigma}_{+}$ and all center geodesics of   M\"obius strips that are crown sets.    Let $U^{\sigma}$ be the   component of $L\sm |R^{\sigma}|$ containing $\Lambda^{\sigma}_{+}$.  Then $R^{\sigma}_{k} = f_{*}^{k}(R^{\sigma})$ is the set consisting of all rims of crowns of $f_{*}^{k}(\Lambda^{\sigma}_{+})$ and all center geodesics of   M\"obius strips that are crown sets of $f_{*}^{k}(\Lambda^{\sigma}_{+})$. Let $U^{\sigma}_{k}$ be the component of $L\sm |R^{\sigma}_{k}|$ containing $f_{*}^{k}(\Lambda^{\sigma}_{+})$.

\begin{figure}[b]
\begin{center}
\begin{picture}(180,100)(60,-100)
\rotatebox{270}{\includegraphics[width=100pt]{CrownSetRight}}

\put (-153,-49){\small$\tilde{\gamma}^{+}_{1}$}
\put (-215,-60){\small$\tilde{\gamma}^{+}_{0}$}
\put (-96,-60){\small$\tilde{\gamma}^{+}_{2}$}
\put (-58,-20){\small$\tilde{\rho}$}
\put (-215,-99){\small$b_{0}$}
\put (-96,-99){\small$b_{2}$}
\put (-153,-99){\small$b_{1}$}
\put (-187,-99){\small$a_{0}$}
\put (-121,-99){\small$a_{1}$}
\put (-120,-29){\small$\tilde{C_{+}}$}
\put (-30,-85){$\dots$}
\put (-286,-85){$\dots$}

\end{picture}
\caption{Lift of crown $C_{+}$ with rim $\rho$}\label{CSright}
\end{center}
\end{figure}

\begin{lemma}\label{minr}

For some minimal $r\ge 0$, $U^{\sigma} = U^{\sigma}_{r}$ and $\Lambda^{\sigma}_{+} = f_{*}^{r}(\Lambda^{\sigma}_{+})$.

\end{lemma}

\begin{proof}

If no crown set of $\Lambda^{\sigma}_{+}$ is a crown, then $L\sm\Lambda^{\sigma}_{+}$ consists of crown sets, so $\Lambda^{\sigma}_{+}\cap f_{*}(\Lambda^{\sigma}_{+})\ne\emptyset$. By Lemma~\ref{disjlams}, $\Lambda^{\sigma}_{+} = f_{*}(\Lambda^{\sigma}_{+})$.

Otherwise, $\Lambda^{\sigma}_{+}$ has at least one crown. First note that if an element of $R^{\sigma}_{i}$ meets an element  of $R^{\sigma}_{j}$ transversely, it follows that  $f_{*}^{i}(\Lambda^{\sigma}_{+})$ meets  $f_{*}^{j}(\Lambda^{\sigma}_{+})$ transversely which contradicts Lemma~\ref{disjlams}. Thus, an element of $R^{\sigma}_{i}$ either coincides with or is disjoint from an element of $R^{\sigma}_{j}$. It follows that the set  $\bigcup_{j=0}^{\infty}R^{\sigma}_{j}$  is finite and there is a smallest $r\ge 0$ so that  $R^{\sigma}_{r }= R^{\sigma}$ and $U^{\sigma}_{r }= U^{\sigma}$. Thus, $\Lambda^{\sigma}_{+}\cap f_{*}^{r}(\Lambda^{\sigma}_{+})\ne\emptyset$. Again by Lemma~\ref{disjlams}, $\Lambda^{\sigma}_{+} = f_{*}^{r}(\Lambda^{\sigma}_{+})$.
\end{proof}

Let $\lambda\in\Lambda^{\sigma}_{+}$ have lift $\wt\lambda$ with endpoints $a_{0}, a_{1}\in\Si$ as in Figure~\ref{b1b2}.  Let $[a_{0}, a_{1}]$ be one of the two intervals on $\Se$ with endpoints $a_{0}, a_{1}$. Let $p>0$ be an integer such that $f_{*}^{p}(\lambda) = \lambda$, $f_{*}^{p}$ preserves the orientation of $\lambda$, and $f^{p}$ is orientation preserving on $L$. Of necessity $p$ is a multiple of $r$. Let $\wt f_{*}^{p}$ be a lift of $f_{*}^{p}$ such that $\wt f_{*}^{p}(\wt\lambda) = \wt\lambda$.

\begin{figure}[h]
\begin{center}
\begin{picture}(200,140)(80,-140)
\rotatebox{270}{\includegraphics[width=145pt]{a0a1b}}

\put (-315,-65){\small$\tilde{\lambda}$}
\put (-250,-50){\small$\zeta$}
\put (-265,-87){\small$\tilde{\lambda}_{0}$}
\put (-355,-72){\small$a_{0}$}
\put (-193,-76){\small$a_{1}$}
\put(-335,-127){\small$A_{0}$}
\put(-350,-94){\small$A$}
\put (-196,-48){\small$B$}
\put(-143,-119){\small$A_{0}$}
\put (-8,-37){\small$B_{0}$}
\put (-175,-76){\small$A=a_{0}$}
\put (3,-67){\small$a_{1}$}
\put (-118,-138){\small$b_{1}$}
\put (-30,-138){\small$b'_{1}$}
\put (-120,-63){\small$\tilde{\lambda}$}
\put (-77,-85){\small$\tilde{\lambda}_{1}$}
\put (-77,-117){\small$\tilde{\lambda}_{2}$}
\put (-60,-45){\small$\tilde{\sigma}$}

\end{picture}
\caption{Lemma~\ref{perleaf}}\label{b1b2}
\end{center}
\end{figure}

\begin{lemma}\label{perleaf}

If $\wt\lambda$ as above is approached by lifts of geodesics in $\Lambda^{\sigma}_{+}$ having endpoints in $(a_{0}, a_{1})\cap\Si$, then,

\begin{enumerate}

\item The points $a_{0}, a_{1}$ are contracting fixed points of $\wh f_{*}^{p}$ in the interval $[a_{0}, a_{1}]$\upn{;}\label{partone}

\item $\wh f_{*}^{p}$ has exactly one fixed point $b_{1}\in(a_{0}, a_{1})\ss\Se$ which is expanding. Further $b_{1}\in\Si$.\label{parttwo}

\end{enumerate}

\end{lemma}

\begin{proof}

Any closed geodesic meeting $U^{\sigma}$ intersects $\Lambda^{\sigma}_{+}$ transversely. Thus $\sigma$ intersects $\lambda$ transversely.  Let $\wt\sigma$ be a lift of $\sigma$ with endpoints $A_{0},B_{0}\in\Si$ with $A_{0}\in(a_{0},a_{1})$ and $B_{0}\notin[a_{0},a_{1}]$. Thus $\wt\sigma\cap\wt\lambda\ne\emptyset$. Let $A_{k} = \wh f_{*}^{kp}(A_{0})$ and $B_{k} = \wh f_{*}^{kp}(B_{0})$. Since $\wh f_{*}^{p}$ is a homeomorphism, it follows that the sequence $\{A_{k}\}_{k\ge 0}$ is monotone in $(a_{0}, a_{1})$ and that $A_{k}\to A\in[a_{0}, a_{1}]\cap\Si$. Similarly $B_{k}\to B\in\Si\sm(a_{0},a_{1})$.  Let  $\zeta$ be the geodesic in $\Delta$ with  endpoints $A,B$. Then $\wh f_{*}^{jp}(\wt\sigma)\to\zeta$ as $j\to\infty$..

Refer to Figure~\ref{b1b2}~(left). If $a_{0}\ne A\ne a_{1}$, by assumption, there exists a lift $\wt\lambda_{0}$ of a leaf of $\Lambda^{\sigma}_{+}$ with one endpoint in the interval $(a_{0},A)\cap\Si$ and the other endpoint in the interval $(A,a_{1})\cap\Si$. Since $\Lambda^{\sigma}_{+}\ss\LL^{\sigma}_{+}$, there exist a subsequence $\{\sigma_{n_{k}p}\}_{k\ge0}$ of the sequence $\{\sigma_{n}\}_{n\ge0}$ with lift  $\{\wt\sigma_{n_{k}p}\}_{k\ge0}$ such that $\wt\sigma_{n_{k}p}\to\wt\lambda_{0}$ as $k\to\infty$. Thus, for arbitarily large values of the integer $k$,  $\wt\sigma_{n_{k}p}$ has   one endpoint in the interval $(a_{0},A)\cap\Si$ and the other endpoint in the interval $(A,a_{1})\cap\Si$. Thus, for arbitrarily large integers $k$, $\wt\sigma_{n_{k}p}\cap\wt f_{*}^{n_{k}p}(\wt\sigma)\ne\emptyset$. Since $\wt\sigma_{n_{k}p}$ and $\wt f_{*}^{n_{k}p}(\wt\sigma)$ are both lifts of $\sigma_{n_{k}p}$, it follows that $\sigma_{n_{k}p}$ is not simple. This contradiction implies that  $A=a_{0}$ or $A=a_{1}$. Without loss assume that $A=a_{0}$. The case $A=a_{1}$ is similar. From the above discussion, the point $a_{0}$ is a contracting fixed point of $\wh f_{*}^{p}$ in the interval $[a_{0}, a_{1}]\ss\Se$ and the only fixed point in the interval $[a_{0}, A_{0})$.

Refer to Figure~\ref{b1b2}~(right).  Choose $\lambda_{1}\in\Lambda^{\sigma}_{+}$ with lift $\wt\lambda_{1}$ having one endpoint in $(a_{0},A_{0})\cap\Si$. Then, $\wt f_{*}^{np}(\wt\lambda_{1})\to \wt\lambda$ as $n\to\infty$  so $a_{1}$ is also a contracting fixed point  of $\wh f_{*}^{p}$ in the interval $[a_{0}, a_{1}]$.

If the endpoints of $\wt f_{*}^{-np}(\wt\lambda_{1})$ converge to $b_{1}\ne b'_{1}\in(a_{0},a_{1})\cap\Si$ as $n\to\infty$, then $b_{1}$ and $b'_{1}$ are both contracting fixed points of $\wh f_{*}^{-p}$ and thus expanding fixed points of $\wh f_{*}^{p}$ in the subinterval $[b_{1},b'_{1}]\ss\Se$ which contains the points $a_{0}, a_{1}$. On the other hand, since $p$ is a multiple of $r$, by Lemma~\ref{minr} it follows that the $\wt f_{*}^{-np}(\wt\lambda_{1})$ are lifts of leaves of $\wt\Lambda^{\sigma}_{+}$ which converge to the lift $\wt\lambda_{2}$ of a leaf of $\Lambda^{\sigma}_{+}$ with endpoints $b_{1},b'_{1}\in\Si$.  Part (\ref{partone}) of this lemma applied to this interval $[b_{1},b'_{1}]$ and the geodesic $\wt\lambda_{2}$ implies that $b_{1}$ and $b'_{1}$ are both  contacting fixed points of $\wh f_{*}^{p}$ in the interval $[b_{1},b'_{1}]$. This contradiction implies $b_{1}=b'_{1}$ and (\ref{parttwo}) follows.
\end{proof}

\begin{figure}[h]
\begin{center}
\begin{picture}(200,140)(50,-140)
\rotatebox{270}{\includegraphics[width=140pt]{1TwoPics}}

\put (-280,-62){\small$\tilde{\lambda}$}
\put (-96,-35){\small$\tilde{\gamma}^{+}_{0}$}
\put (-55,-35){\small$\tilde{\gamma}^{+}_{2}$}
\put (-81,-58){\small$\tilde{\lambda} = \tilde{\gamma}^{+}_{1}$}
\put (-75,-20){\small$\wt C_{+}$}
\put (-132,-30){\small$b_{0}$}
\put (-15,-28){\small$b_{2}$}
\put (-245,-2){\small$b_{0}$}
\put (-243,-144){\small$b_{1}$}
\put (-69,-142){\small$b_{1}$}
\put (-317,-70){\small$a_{0}$}
\put (-173,-70){\small$a_{1}$}
\put (-143,-68){\small$a_{0}$}
\put (0,-70){\small$a_{1}$}
\put (-229,-42){\small$\wt\gamma$}
\put (-87,-95){\small$\wt\gamma^{-}_{1}$}
\put (-58,-95){\small$\wt\gamma^{-}_{2}$}

\end{picture}
\caption{Two Cases}\label{TwoPics}
\end{center}

\end{figure}

Further,

\begin{prop}\label{bothsides}

If $\wt\lambda$ is as in \emph{Lemma~\ref{perleaf}}, then either,

\begin{enumerate}

\item $\wt\lambda$ is appoached from above and below by leaves of $\wt\Lambda^{\sigma}_{+}$ and there exist exactly four fixed points $a_{0},a_{1},b_{0},b_{1}$ of $\wh f_{*}^{p}$ in $\Se$  as in \emph{Figure~\ref{TwoPics} (left)}\upn{;}\label{prtone}

\item $\wt\lambda$ is bordered above by the lift of a crown set $C_{+}$ as in \emph{Figure~\ref{TwoPics} (right)} and $\lambda = \gamma^{+}_{1}$ belongs to a cycle $\{\gamma^{+}_{i}\}_{i\in\mathbb{Z}}\ss\delta C_{+}$ with lifts   $\{\wt\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$  such that, for $i\in\mathbb{Z}$,  \label{prttwo}

\begin{enumerate}

\item $\wt\gamma^{+}_{i}$ and $\wt\gamma^{+}_{i+1}$ share endpoint $a_{i}\in\Si$\upn{;}

\item $a_{i}$ is a contracting fixed point of $\wh f_{*}^{p}$ in $\Se$\upn{;}\label{part2}

\item There exists $b_{i}\in(a_{i-1},a_{i})\cap\Si$ which is an expanding fixed point of $\wh f_{*}^{p}$ in $\Se$ and the only fixed point of $\wh f_{*}^{p}$ in $(a_{i-1},a_{i})$.

\end{enumerate}

\end{enumerate}

\end{prop}

\begin{proof}

If $\wt\lambda$ is approached both above and below (in Figure~\ref{b1b2}~(right)) by lifts of leaves of $\Lambda^{\sigma}_{+}$, then (\ref{prtone}) follows by applying Lemma~\ref{perleaf}~(\ref{partone}) to each of the intervals $[a_{0},a_{1}]\ss\Se$.

Otherwise, there exists a component $V$ of $L\sm|\Lambda^{\sigma}_{+}|$ such that $\lambda\in\delta V$. Thus, there exists a crown set $C_{+}$ and cycle of geodesics $\{\gamma^{+}_{i}\}_{i\in\mathbb{Z}}\ss\delta C_{+}$ such that  $\lambda = \gamma^{+}_{1}$ belongs to the cycle. The lift $\wt\lambda$ of $\lambda$ fixed by $\wt f_{*}^{p}$ determines lifts $\wt\gamma^{+}_{i}$ of $\gamma^{+}_{i}$, $i\in\mathbb{Z}$, all fixed by $\wt f_{*}^{p}$. Then (\ref{prttwo})  follows from applying Lemma~\ref{perleaf}~(\ref{parttwo}) to each of the geodesics $\wt\gamma^{+}_{i}$, $i\in\mathbb{Z}$.
\end{proof}

\begin{rem}

We will see that the pairs of points  $b_{i-1},b_{i}\in\Si$, $i\in\mathbb{Z}$, are each endpoints of leaves of $\wt\Lambda^{\sigma}_{-}$

\end{rem}

\begin{example}

Proposition~\ref{bothsides}~(\ref{prttwo}) is illustrated in Figure~\ref{CSleft} if the crown set $C_{+}$ is simply connected, in Figure~\ref{CScusp} if the crown set is a cusp, and Figure~\ref{CSright} if the crown set is an annulus or a M\"obius strip.

\end{example}

\begin{cor}\label{trtau}

If $\tau$ is a simple closed geodesic transverse to $\Lambda^{\sigma}_{+}$, then $\tau_{n} = f_{*}^{n}(\tau)$, $n\in\mathbb{Z}$, are distinct.

\end{cor}

\begin{proof}

Let $\wt\tau$ be a lift of $\tau$ meeting $\wt\lambda$ at a point close to $a_{0}\in\Si$. If $f_{*}^{k}(\tau) = \tau$, then $\wt f_{*}^{-nkp}(\wt\tau)$, $n\ge 0$, is a sequence of lifts of $\tau$ and as such should not accumulate in $\Delta$. One can see from Figure~\ref{b1b2} that  this sequence  limits on the geodesic with endpoints $b_{0},b_{1}\in\Si$. The corollary follows from this contradiction.
\end{proof}

\begin{cor}

If $\tau$ is a simple closed geodesic  meeting $|R^{\sigma}|$, then $\tau_{n} = f_{*}^{n}(\tau)$, $n\in\mathbb{Z}$, are distinct.

\end{cor}

\begin{proof}

If $\tau$ meets $|R^{\sigma}|$, then $\tau$ is transverse to $\Lambda^{\sigma}_{+}$ and the corollary follows from the previous corollary.
\end{proof}

The previous two corollaries have as immediate consequence.

\begin{cor}\label{corcomp}\hfill

\begin{enumerate}

\item The elements of $R^{\sigma}$ not components of $\bd L$ are contained in $\Gamma$\upn{;}

\item If $U^{\sigma}$ is the component  of $L\sm|R^{\sigma}|$ containing $\Lambda^{\sigma}_{+}$, then $U^{\sigma} = U$\upn{;}

\item $U\sm\Lambda^{\sigma}_{+}$ is a finite union of sets whose closures are crown sets\upn{;}\label{itemcompl}

\item If $\tau\ss U\sm\bd L$ is a simple closed  geodesic, then   $\tau_{n} = f_{*}^{n}(\tau)$, $n\in\mathbb{Z}$, are distinct.

\end{enumerate}

\end{cor}

\subsection{The negative  lamination $\Lambda^{\sigma}_{-}$}\label{neglams}

 We define the negative lamination $\Lambda^{\sigma}_{-}$ and develop its properties exactly as in Sections~\ref{poslam} and~\ref{crsets}.  All the results of Sections~\ref{poslam} and~\ref{crsets}  hold for   $\Lambda^{\sigma}_{-}$   with the obvious modifications.

Proposition~\ref{neglam} below relates $\Lambda^{\sigma}_{-}$ and $\Lambda^{\sigma}_{+}$. We need two preliminary lemmas.

\begin{lemma}\label{pmtrans}

If $\lambda_{-}\in\Lambda^{\sigma}_{-}$ and $\lambda_{+}\in\Lambda^{\sigma}_{+}$ are  semi-isolated leaves, then $\lambda_{-}\cap\lambda_{+}$ is nonempty and transverse.

\end{lemma}

\begin{proof}

Since the components of $U\sm\Lambda_{+}$  are open disks, open or half-open annuli, or open M\"obius strips, it follows that $\lambda_{-}\cap\lambda_{+}$ is nonempty. It remains to show that $\lambda_{-}\ne\lambda_{+}$. Suppose $\lambda_{-} = \lambda_{+} = \lambda$ and choose the integer $p>0$ and lifts $\wt\lambda$ of $\lambda$ and $\wt f_{*}^{p}$ of $f_{*}^{p}$  so that $\wt f_{*}^{ p}(\wt\lambda) = \wt\lambda$ and $\wt f_{*}^{p}$ preserves the orientation of $\wt L$ and $\wt\lambda$. Refer to Figure~\ref{TwoPics}~(right). Then Proposition~\ref{bothsides}~(\ref{prttwo}) applied to $\Lambda^{\sigma}_{+}$ implies that the endpoints $a_{0},a_{1}\in\Si$ of $\wt\lambda$ are contracting fixed points of $\wt f_{*}^{p}$ while Proposition~\ref{bothsides}~(\ref{prttwo}) applied to $\Lambda^{\sigma}_{-}$ implies that the endpoints $a_{0},a_{1}\in\Si$ of $\wt\lambda$ are expanding fixed points of $\wt f_{*}^{p}$. This contradiciton implies $\lambda_{-}\ne\lambda_{+}$.
\end{proof}

\begin{lemma}\label{bothends}

If  $\lambda\in\Lambda^{\sigma}_{\pm}$  and $\gamma$ is a geodesic meeting $\lambda$ transversely, then the points $\lambda\cap\gamma$ accumulate at   both ends of $\lambda$.

\end{lemma}

\begin{proof}

The lemma follows immediately since  $\Lambda^{\sigma}_{\pm}$ are minimal laminations.
\end{proof}

\begin{prop}\label{neglam}

\begin{enumerate}

\item In \emph{Proposition~\ref{bothsides}~(\ref{prtone})} applied to $\wt\Lambda^{\sigma}_{+}$, there exists a minimal lamination $\Lambda\ss\LL^{\sigma}_{-}$ such that the geodesic $\wt\gamma$ with endpoints $b_{0},b_{1}\in\Si$, is the lift of  a   leaf  $\gamma\in\Lambda$ that is not semi-isolated in $\Lambda$\upn{;}\label{itemone}

\item In \emph{Proposition~\ref{bothsides}~(\ref{prttwo})} applied to $\wt\Lambda^{\sigma}_{+}$, there exists a minimal lamination $\Lambda\ss\LL^{\sigma}_{-}$ such that  the geodesics $\wt\gamma^{-}_{i}$ with endpoints $b_{i-1},b_{i}\in\Si$, are the lifts of  semi-isolated leaves   $\gamma^{-}_{i}\in\Lambda$ $i\in\mathbb{Z}$.\label{itemtwo}

\end{enumerate}

\end{prop}

\begin{proof}

(\ref{itemone}) Refer to  Figure~\ref{TwoPics}~(left). Let $\lambda\in\Lambda^{\sigma}_{+}$ with lift $\wt\lambda\in\wt\Lambda^{\sigma}_{+}$ be the leaf of Proposition~\ref{bothsides}~(\ref{prtone}).  Denote by $\wt\gamma$ the geodesic in $\Delta$    having endpoints $b_{0},b_{1}\in\Si$.

By Lemma~\ref{bothends}, lifts of $\sigma$ approach both ends of $\wt\lambda$ so there exists a lift $\wt\sigma$ of $\sigma$ meeting $\wt\lambda$ to the left of $\wt\gamma$ in Figure~\ref{TwoPics}~(left).  Thus, $\wt\sigma_{kp} = \wt f_{*}^{kp}(\wt\sigma)\to\wt\gamma$ from the left as $k\to-\infty$. It follows that $\wt\gamma$ is the lift of a simple geodesic $\gamma\ss L$.  Let $q>0$ be an integer and let $n_{k} = -kp$. Since $\{\wt\sigma_{n_{k}q}\}_{k\ge0}$ is a subsequence of $\{\wt\sigma_{kp}\}_{k\le0}$, it follows that $\wt\sigma_{n_{k}q}\to\wt\gamma$ as $k\to\infty$. By Definition~\ref{defnLL},  $\gamma\in\LL^{\sigma}_{-}$.

Let $\Lambda$ be the lamination consisting of the geodesics in $\ol\gamma$. By Lemma~\ref{bothends}, if $\gamma'\in\Lambda$, then there exists a lift $\wt\gamma'$ of $\gamma'$ meeting $\wt\lambda$ to the left of $\wt\gamma$ in Figure~\ref{TwoPics}~(left). Thus, $\wt f_{*}^{kp}(\wt\gamma')\to\gamma$ from the left as $k\to-\infty$. Similarly, there exists a lift $\wt\gamma''$ of $\gamma'$ such that $\wt f_{*}^{kp}(\wt\gamma''_{-})\to\wt\gamma$ from the right as $k\to-\infty$. It follows that $\gamma$ is not semi-isolated in $\Lambda$ and that $\Lambda$ is a minimal lamination. By the previous paragraph $\Lambda\ss\LL^{\sigma}_{-}$  and (\ref{itemone}) is proven.

(\ref{itemtwo}) is proven similarly. Refer to Figures~\ref{TwoPics}~(right), \ref{CSleft}, \ref{CScusp}, and~\ref{CSright}.
\end{proof}

We take $\Lambda$ of Proposition~\ref{neglam}~(\ref{itemtwo}) as $\Lambda^{\sigma}_{-}$.

\begin{cor}\label{trlams}

$\Lambda^{\sigma}_{+}$ and $\Lambda^{\sigma}_{-}$ are transverse.

\end{cor}

\begin{proof}

Refer to Figures~\ref{TwoPics}~(right). The geodesic $\wt\gamma^{-}_{1}$ with endpoints $b_{0},b_{1}\in\Si$ is a semi-isolated leaf of $\wt\Lambda^{\sigma}_{-}$ which is transverse to $\wt\Lambda^{\sigma}_{+}$. Since $\Lambda^{\sigma}_{-}$ is minimal, if $\gamma^{-}_{1}$ is the leaf of $\Lambda^{\sigma}_{-}$ with lift $\wt\gamma^{-}_{1}$, then $\gamma^{-}_{1}$ is dense in  $\Lambda^{\sigma}_{-}$. It follows that $\Lambda^{\sigma}_{-}$ is transverse to $\Lambda^{\sigma}_{+}$.
\end{proof}

Suppose $C_{+}$ is the  crown set of Proposition~\ref{bothsides}~(\ref{prttwo}) associated to $\{\gamma^{+}_{i}\}_{i\in\mathbb{Z}}\ss\Lambda^{\sigma}_{+}$  Let $p>0$, $\wt f_{*}^{p}$ be a lift of $f_{*}^{p}$, $\{\wt\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$ be a lift of $\{\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$, and $b_{i}$, $i\in\mathbb{Z}$, be fixed points of $\wt f_{*}^{p}$ as in Proposition~\ref{bothsides}~(\ref{prttwo}). Then, if $\gamma^{-}_{i}$ is the geodesic in $L$ with lift $\wt\gamma^{-}_{i}$ having endpoints $b_{i-1},b_{i}\in\Si$,  $i\in\mathbb{Z}$, then let $C_{-}$ be the crown set associated to $\{\wt\gamma^{-}_{i}\}_{i\in\mathbb{Z}}\ss\Lambda^{\sigma}_{-}$.

\begin{defn}

The crown sets $C_{\pm}$ of $\Lambda_{\pm}$ are said to be  \emph{complementary}.

\end{defn}

The closure of components of $U\sm(|\Lambda^{\sigma}_{+}|\cup|\Lambda^{\sigma}_{-}|)$ are components of the intersection of (not necessarily complementary) crown sets  $C_{\pm}$ of $\Lambda^{\sigma}_{\pm}$. All but finitely many of these components  are rectangles.  The nonrectangular components consist of one component of the intersection of some pair of  complementary crown sets.

\begin{defn}

The nonrectangular component of $C_{+}\cap C_{-}$ is called the \emph{nucleus} of the complementary crown sets $C_{\pm}$.

\end{defn}

\begin{rem}

The nulcleus of complemtary crown sets $C_{\pm}$ has one boundary component $\zeta$ which is piecewise geodesic closed curve with edges altenately in  $\gamma^{+}_{i}$ and $\gamma^{-}_{i+1}$, $i\in\mathbb{Z}$. Additionally the nucleus of complementary crowns $C_{\pm}$ will have a boundary component that is the common rim of $C_{\pm}$.

\end{rem}

\begin{cor}\label{compcompl}

The finitely many nuclei  of complementary crown sets $C_{\pm}$ of $\Lambda^{\sigma}_{\pm}$   are either,

\begin{enumerate}

\item Closed disks\upn{;}

\item Punctured disks\upn{;}

\item Annuli with one edge the common rim of the complementary crowns  $C_{\pm}$\upn{;}

\item M\"obius strips.

\end{enumerate}

\end{cor}

\subsection{Further properties of $\Lambda^{\sigma}_{\pm}$ and $\LL^{\sigma}_{+}$}

\begin{prop}\label{lamuniq}

The laminations $\Lambda^{\sigma}_{\pm}$ are independent of the choice of  minimal lamination $\Lambda^{\sigma}_{\pm}\ss\LL^{\sigma}_{\pm}$, and simple closed geodesic $\sigma\ss U$ such that $f_{*}^{n}(\sigma)$, $n\in\mathbb{Z}$, are distinct.

\end{prop}

\begin{proof}

We prove the proposition for $\Lambda^{\sigma}_{+}$. The other case is analogous. Suppose $\Lambda\ss\LL^{\tau}_{+}$ is a minimal lamination of   $\LL^{\tau}_{+}$ and $\tau\ss U$ a simple closed geodesic   such that $f_{*}^{n}(\tau)$, $n\in\mathbb{Z}$, are distinct. If $\gamma\in\Lambda$, since $\Lambda^{\sigma}_{+}$ is minimal and $\Lambda\ss U$, then $\gamma\cap\gamma^{-}_{1}$ clusters at both ends of $\gamma^{-}_{1}$ where $\gamma^{-}_{1}$ is the leaf of $\Lambda^{\sigma}_{-}$ with lift $\wt\gamma^{-}_{1}$ having endpoints  $b_{0},b_{1}\in\Si$ (Figure~\ref{TwoPics}~(right)). Choose a lift $\wt\gamma$  of $\gamma$ near $b_{1}$. Then $\wt f_{*}^{np}(\wt\gamma)\to\wt\gamma^{+}_{1}$ as $n\to\infty$. Thus $\gamma^{+}_{1}\in\Lambda$ so $\Lambda^{\sigma}_{+}\ss\Lambda$. Since $\Lambda$ is minimal, it follows that $\Lambda^{\sigma}_{+} = \Lambda$.
\end{proof}

\begin{defn}

Define $\Lambda_{\pm} = \Lambda^{\sigma}_{\pm}$.

\end{defn}

\begin{rem}

We will still use the  notation $\Lambda^{\sigma}_{\pm}$ because it specifies the geodesic $\sigma$.

\end{rem}

\begin{lemma}\label{finitemany}

$\LL^{\sigma}_{+}\sm\Lambda^{\sigma}_{+}$ and $\LL^{\sigma}_{-}\sm\Lambda^{\sigma}_{-}$ are each a finite set of geodesics.

\end{lemma}

\begin{proof}

We prove that $\LL^{\sigma}_{+}\sm\Lambda^{\sigma}_{+}$ is finite. The proof of the other case is analogous. We use the notation of Proposition~\ref{bothsides}~(\ref{prttwo}). Thus, if $\gamma\in\LL^{\sigma}_{+}\sm\Lambda^{\sigma}_{+}$, then $\gamma$ lies in a crown set $C_{+}$. It follows that the lift $\wt\gamma$ of $\gamma$ to $\wt C_{+}$ has vertices $a_{i}, a_{j}\in\Si$, $i\ne j\in\mathbb{Z}$. Since the set $\{\gamma^{+}_{i}\}_{i\in\mathbb{Z}}$ is finite, there can be only finitely many such $\gamma$.
\end{proof}

The proof of Theorem~\ref{proplams} is complete.

\section{Defining $h$ and the proof of Theorem~\ref{NTthm}}

By~\cite[Theorems~2.1 and~3.3]{Epstein:isotopy}  and induction,  there exists an ambient isotopy $\Phi:L\times[0,1]\to L$ such that $\Phi^{0} = \id$ and $\Phi^{1}\circ f(\gamma) = f_{*}(\gamma)$ for every   $\gamma\in\Gamma$. Here we use the notation, $\Phi^{t}(x) = \Phi(x,t)$. In abuse of notation, denote $\Phi^{1}\circ f$ by $f$.  Thus, from now on we  assume  that $f(\gamma) = f_{*}(\gamma)$  for every   $\gamma\in\Gamma$.

\begin{prop}

There exists a homeomorphism $h:L\to L$, isotopic to $f$such that,

\begin{enumerate}

\item If $U$ is  a component of $L\sm|\Gamma|$, then $h(U)$ is  a component of $L\sm|\Gamma|$\upn{;}\label{111}

\item If $U$ is a periodic (respectively pseudo-Anosov) component, then $h(U)$  is a periodic (respectively pseudo-Anosov) component\upn{;}\label{222}

\item $h(\Lambda^{\sigma}_{\pm}) = \Lambda^{f_{*}(\sigma)}_{\pm}$ if $\sigma$ is a simple closed  geodesic in a pseudo-Anosov component.

\end{enumerate}

\end{prop}

\begin{proof}

Choose lifts $\wt L$ af $L$ and $\wt f:\ol L\to\wt L$ of $f:L\to L$. We will define $\wt h:\wt L\to\wt L$ so that $\wh h|\Se.= \wh f|\Se$.  First define $\wt h|\wt\gamma = \wt f|\wt\gamma$ if $\gamma\in\Gamma$ or $\gamma\ss\bd L$. Then (\ref{111}) follows since $f(\gamma) = f_{*}(\gamma)$  for every   $\gamma\in\Gamma$. Further,  if $U$ is a pseudo-Anosov somponent and $\sigma\ss U\sm\bd L$ is a simple closed  geodesic, then Lemma~\ref{f*seq} implies that  $f_{*}(\Lambda^{\sigma}_{\pm}) = \Lambda^{f_{*}(\sigma)}_{\pm}$ and $f(U)$ is a pseudo-Anosov component     and  (\ref{222}) follows.

Next define  $\wt h|\wt U = \wt f|\wt U$ if $U$ is a periodic component.

Now suppose $U$ is a pseudo-Anosov component and $\sigma\ss U$ is a simple closed  geodesic. To define $\wt h:\wt U\to\wt{h(U)}$, we mimic the proof of Lemma~6.1 of Casson-Bleiler~\cite[pp.~89-90]{bca}.   Choose  lifts $\wt U$ of $U$, $\wt{h(U)}$ of $h(U)$, and  $\wt f:\wt U\to \wt{h(U)}$ of $f$.  Let $\wh U$ (respectively $\wh{h(U}$) be the closure of $\wt U$ (respectively $\wt{h(U}$)  in $\mathbb{D}^{2}$  and let $\wh f:\wh U\to\wh{h(U)}$ be the extension of $\wt f$ to $\wh U$.  Let
$$X = |\Lambda^{\sigma}_{+}|\cap|\Lambda^{\sigma}_{-}|, \qquad X^{*} = |\Lambda^{f_{*}(\sigma)}_{+}|\cap|\Lambda^{f_{*}(\sigma)}_{-}|$$  and $\wt X$ (respectively $\wt X^{*}$) the  lift of   $X$ to $\wt U$ (respectively $X^{*}$ to $\wt{h(U)}$). Define $\wt h:\wt X\to\wt X^{*}$ as follows. If $\wt x = \wt \lambda_{-}\cap\wt\lambda_{+}$ with $\wt\lambda_{-}\in\wt\Lambda_{-}$ and $\wt\lambda_{+}\in\wt\Lambda_{+}$, define $\wt h(\wt x) = \wt f_{*}(\wt\lambda_{-})\cap\wt f_{*}(\wt\lambda_{+})$.

The fact that $\wh f:\wh U\to\wh{h(U)}$ is continuous implies that the map $\wt h:\wt X\to\wt X^{*}$ is a homeomorphism.    As in   Casson-Bleiler~\cite[pp.~89-90]{bca}, we extend $\wt h$ linearly and equivariantly over any lift of an arc of $|\wt\Lambda^{\sigma}_{+}|\cup|\wt\Lambda^{\sigma}_{-}|$ with both endpoints in $\wt X$ and interior disjoint from $\wt X$ and equivariantly over the  lifts of rectangular components of $U\sm(|\Lambda^{\sigma}_{+}|\cup|\Lambda^{\sigma}_{-}|)$   using the technique of  Casson-Bleiler~\cite[pp.~90]{bca}. It follows from Corollary~\ref{compcompl} that the closures of the  nonrectangular components of $U\sm(|\Lambda^{\sigma}_{+}|\cup|\Lambda^{\sigma}_{-}|)$ consist of finitely many sets which are either,
\begin{enumerate}

\item Annuli that are the nuclei of pairs of complementary crowns with one edge a rim of the crown\upn{;}\label{first}

\item Disks that are the nuclei of pairs of complementary crown sets\upn{;}

\item  Punctured disks that are the nuclei of pairs of complementary crown sets\upn{;}

\item M\"obius strips that are the nuclei of pairs of complementary crown sets\upn{;}\label{fourth}

\item Annuli formed by pasting   the nuclei of two pairs of complementary crowns   together along their common rims.\label{fifth}

\end{enumerate}

The rims of the crown sets in cases~(\ref{first}) and~(\ref{fifth}) and the center geodesic of the M\"obius strips in case~(\ref{fourth}) belong to $\Gamma$  so the homeomorphism $h$ has already been defined above on these geodesics. In all cases, the homeomorphism $\wt h$ can be extended equivariantly in an  arbitrary way over the lifts of the nonrectangular components of $U\sm(|\Lambda^{\sigma}_{+}|\cup|\Lambda^{\sigma}_{-}|)$.

Both $\wh f,\wh h:\wh U\to \wh{h(U)}$ are defined and agree on $\Se$. Thus,~\cite[Corollary~5]{cc:epstein} implies that the homeomorphisms $h$ and $f$ are isotopic.
\end{proof}

\subsection{Completing the proof of Theorem~\ref{NTthm}}

The next two lemmas complete the proof of Theorem~\ref{NTthm}. In the lemmas, $U$ is a component of $L\sm|\Gamma|$, $S$ is the internal completion of $U$, and $n_{S} = n_{U}$. The map $h:U\to h(U)$ extends in a natural way to a map $h:S\to h(S)$ where $h(S)$ is the internal completion of $h(U)$.

\begin{lemma}

If $S$ is a pseudo-Anosov component, the homeomorphism $h^{n_{S}}:S\to S$ satisfies \emph{Theorem~\ref{NTthm}~(\ref{NT2})}.

\end{lemma}

\begin{proof}

Since $h:U\to h(U)$ satisfies $h(\Lambda^{\sigma}_{\pm}) = \Lambda^{f_{*}(\sigma)}_{\pm}$, it follows that $h:S\to h(S)$ satisfies $h(\Lambda^{\sigma}_{\pm}) = \Lambda^{f_{*}(\sigma)}_{\pm}$ and $h^{n_{S}}:S\to h^{n_{S}}(S) = S$ satisfies $h^{n_{S}}(\Lambda^{\sigma}_{\pm}) = \Lambda^{f^{n_{S}}_{*}(\sigma)}_{\pm}$.  But by Proposition~\ref{lamuniq}, $\Lambda^{f^{n_{S}}_{*}(\sigma)}_{\pm} = \Lambda^{\sigma}_{\pm}$.
\end{proof}

\begin{lemma}

If $S$ is a periodic component, the homeomorphism $h^{n_{S}}:S\to S$ satisfies \emph{Theorem~\ref{NTthm}~(\ref{NT1})}.

\end{lemma}

\begin{proof}[Sketch of proof]

First assume that $S$ is not planar with $b+c=3$.  It is elementary to construct two finite sets $\Sigma$ and $\Sigma'$ of pairwise disjoint, simple closed geodesics in $S$ such that the components of $S\sm(|\Sigma|\cup|\Sigma'|)$ consist of finitely many peripheral annuli, finitely many annular neighborhoods of cusps, and finitely many open disks.

Since $S$ is a periodic component, for each $\sigma\in\Sigma\cup\Sigma'$, there exists an integer $k>0$ such that $\sigma_{kn_{S}} = \sigma$. Thus, there exists an integer $n>0$ such that $\sigma_{nn_{S}} = \sigma$, all $\sigma\in\Sigma\cup\Sigma'$.  Denote $(h^{n_{S}}|S)^{n}$ by $g$.  Explicity construct an ambient  isotopy $\Phi:L\times[0,5]\to L$ such that $\Phi^{0} = \id$ and $\Phi^{5}\circ g = \id$ as follows,

\begin{enumerate}

\item Use~\cite[Theorems~2.1 and~3.3]{Epstein:isotopy}  and induction to  define $\Phi^{t}$, $t\in[0,1]$, moving $g(\sigma)$ to $\sigma$ for every   $\sigma\in\Sigma$\upn{;}

\item Use the techniques of~\cite[Lemmas~4.58 and~4.60]{cc:hm}, to define $\Phi^{t}$, $t\in[1,2]$,   moving $\Phi^{1}\circ g(\sigma)$ to $\sigma$ for every   $\sigma\in\Sigma'$   while keeping each    $\sigma\in\Sigma$ fixed\upn{;}

\item It is elementary to define $\Phi^{t}$,  $t\in[2,3]$, supported in a neighborhood of $|\Sigma|\cup|\Sigma'|$ and fixing $|\Sigma|\cup|\Sigma'|$, such that   $\Phi^{3}\circ g = \id$ on $|\Sigma|\cup|\Sigma'|$\upn{;}

\item Use Alexander's trick to define $\Phi^{t}$, $t\in[3,4]$, so that $\Phi^{4}\circ g = \id$ on the simple connected components of $S\sm(|\Sigma|\cup|\Sigma'|)$\upn{;}

\item It is elementary to define $\Phi^{t}$,  $t\in[4,5]$, supported on the annular components of $S\sm(|\Sigma|\cup|\Sigma'|)$, so that $\Phi^{5}\circ g = \id$ on the annilar components of $S\sm(|\Sigma|\cup|\Sigma'|)$.

\end{enumerate}
Thus, $\Phi^{0}\circ (h^{n_{S}}|S)^{n} = (h^{n_{S}}|S)^{n} $ and $\Phi^{5}\circ (h^{n_{S}}|S)^{n} = \id$.

If $S$ is planar with $b+c=3$, the lemma is routine.
\end{proof}

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redpajama_set_name: RedPajamaArXiv


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Prof. Chao Wang
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Area of Expertise: Microwave photonics, Photonic signal processing, Optical sensors, Ultrafast imaging, Fiber Bragg grating
Chao Wang is currently a Lecturer (Assistant Professor) in the School of Engineering and Digital Arts,, University of Kent, UK. His research interests lie in the fields of microwave photonics, ultrafast imaging, optical sensing and biophotonics for biomedical and defense applications.
Prior to joining Kent, Chao was a NSERC Postdoctoral Fellow in the Photonics Lab at University of California, Los Angeles (UCLA), USA. He received his BEng degree in Opto-electronic Engineering from Tianjin University,China, in 2002, his MASc degree in Optics from Nankai University, China, in 2005 and his PhD degree in Electrical and Computer Engineering from the University of Ottawa, Canada, in 2011.
He has been granted several international and regional awards, including SPIE Scholarship in Optical Science and Engineering (2008), Vanier Canada Graduate Scholarship (2009), Chinese Government Award for Outstanding Self-Financed Students Abroad (2009), IEEE Photonics Society Graduate Student Fellowship (2009), IEEE Microwave Theory and Techniques Society Graduate Fellowship (2010), NSERC Postdoctoral Fellowship (2011) and Marie Curie Career Integration Grant Award (2014).
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Through a single view, you can assess your assets, service requests and work orders while combining basemap data with changing, real-time event data from cameras and sensors that enable you to edit and update key information.
Budget forecasting, facility inventory analysis, customer billing and key administrative functions can be enhanced through the implementation of an enterprise-wide GIS solution.
GIS spatial selection and display tools allow users to visualize scheduled work, ongoing activities, recurring maintenance problems and historical information.
GIS provides you with an intelligent database so that job planning, equipment inventory and flow analysis become an automated procedure integrated into one system.


text:
<?php
namespace Index\Model;

 use Zend\Db\TableGateway\TableGateway;

 class articleTable
 {
     protected $tableGateway;

     public function __construct(TableGateway $tableGateway)
     {
         $this->tableGateway = $tableGateway;
     }

     public function fetchAll($where='',$order='',$limit='')
     {

            $resultSet = $this->tableGateway->select($where,$order,$limit);

         return $resultSet;
     }

     public function getartCat($id)
     {
         $id  = (int) $id;
         $rowset = $this->tableGateway->select(array('artCat_id' => $id));
         $row = $rowset->current();
         if (!$row) {
             throw new \Exception("Could not find row $id");
         }
         return $row;
     }

     public function saveBanner(Banner $album)
     {
         $data = array(
                'banner_id'      => $this->banner_id,
                'banner_title'   => $this->banner_title,
                'banner_url'     => $this->banner_url,
                'banner_image'   => $this->banner_image,
                'banner_show'    => $this->banner_show,
                'banner_orderBy' => $this->banner_orderBy,
         );

         $id = (int) $album->id;
         if ($id == 0) {
             $this->tableGateway->insert($data);
         } else {
             if ($this->getAlbum($id)) {
                 $this->tableGateway->update($data, array('banner_id' => $id));
             } else {
                 throw new \Exception('Banner id does not exist');
             }
         }
     }

     public function deleteBanner($id)
     {
         $this->tableGateway->delete(array('banner_id' => (int) $id));
     }

     function checksql(){
         return $this->tableGateway->getLastInsertValue();
     }
 }
?>

redpajama_set_name: RedPajamaGithub


Nintendo's general manager for the U.K. and the company's marketing manage believe the "heyday of piracy" in the games business is past, according to CVG report.
Has game piracy hit its plateau? Nintendo marketing manager for the U.K., James Honeywell certainly believes so.
"I think perhaps there's been a 'heyday of piracy' and we've now seen a lot of rules come in to stop it," Honeywell told gaming publication CVG in an interview published today.
Honeywell noted that piracy has been a drain on economies around the world, and countries are starting to realize that by limiting piracy, they can benefit financially.
"There's definitely a step change coming and you see it in various countries around the world," Honeywell told CVG. "People are aware that video games, music, and movies make massive contributions to the economies of countries. They need to make sure they start protecting those things."
Nintendo is also doing its part. The company's 3DS portable-gaming device, which is set tolaunch in the U.S. on March 27 for $249, is expected to have extremely strong piracy countermeasures in place.
"We can't divulge any technical details on [3DS anti-piracy controls]," Nintendo general manager for the U.K. David Yamton told CVG, "but needless to say this is probably one of our best pieces of equipment in that respect. There are a lot of things we've learned over time to try and improve the security and protection--not only of our IP but of our third-party publishers' IP as well."
Over the past few months, Nintendo has been quite outspoken about its fight against piracy.
In an earnings call with investors in October, Nintendo Chief Executive Satoru Iwata said that his company has a "responsibility" to fight piracy. He specifically mentioned the company's 3DS and its "countermeasures" as key aspects of that battle.
So far, that battle is one that Nintendo has not enjoyed. In 2009, the company's New Super Mario Bros. Wii title was illegally downloaded 1.15 million times, according to data from TorrentFreak. Punch-Out and Wii Sports Resort were also heavily pirated with 950,000 and 920,000 downloads, respectively.
At the end of 2010, things weren't going any better for Nintendo. TorrentFreak revealed last month that Super Mario Galaxy 2 was pirated 1.47 million times last year. Wii Party was downloaded 1.22 million times. Donkey Kong Country Returns rounded out the top three with 920,000 pirated copies.
Call of Duty: Black Ops was the most pirated game of 2010. The PC version alone was illegally downloaded more than 4.27 million times.
Given those figures, Honeywell's claim that piracy is getting past its prime might be a little premature.


text:
DLL_PUBLIC(GUI)

add_definitions(-DNO_INPUT=1)

# include
include_directories(AFTER ${M3E_SOURCE_DIR}/src/frameworks/native/libs/gui/include)


text:
\section{Introduction}
Pursuing room-temperature superconductors has been a major theme in condensed-matter and materials physics since the discovery of superconductivity in 1911.
According to the Bardeen–Cooper–Schrieffer (BCS) theory,
strong electron-phonon coupling and high phonon frequencies can conspire to produce superconductivity exhibiting high critical temperatures ($T_c$'s).
These two conditions can be achieved via strong covalent metallicity and low atomic mass, respectively.
Naturally, metallic hydrogen and hydrogen-rich materials in general under pressure are plausible candidate high-$T_c$ superconductors~\cite{Ashcroft1968,Ashcroft2004,Tse2007,Chen2008,Eremets2008,Li2010,Gao2010,Zha2012,Wang2012,Feng2006}.
As a prime example, hydrogen sulfide H$_3$S with the $Im\bar{3}m$ structure has been theoretically predicted~\cite{Li2014,Duan2015} and
experimentally confirmed~\cite{Drozdov2015,Einaga2016} to exhibit a maximum $T_c$ of $203$~K at $150$~GPa.
{\color{black}Though a range of $T_c$'s has been reported in various studies~\cite{Drozdov2015,Einaga2016,Troyan2016,Huang2019,Shimizu2020,Mao2018},
the maximum $T_c$ of at least $180$~K for H$_3$S has now been reproduced in several different experiments~\cite{Drozdov2015,Einaga2016,Shimizu2020}.}
This breakthrough has attracted a great deal of research interest~\cite{Bernstein2015,Errea2015,Papaconstantopoulos2015,Akashi2015,Nicol2015,Heil2015}.
In particular, it has been predicted that under hole doping the $T_c$ of H$_3$S$_{0.927}$P$_{0.075}$ can be as high as $280$~K at $250$~GPa~\cite{Ge2016}.
To date a variety of new hydrogen-rich materials with different structures have been proposed based on first-principles calculations~\cite{Peng2017,Liu2017,Tanaka2017,Ye2018,Cui2020,Sun2020}.
As a paradigmatic system, the lanthanum superhydride LaH$_{10}$ with its novel hydrogen clathrate structure
has been experimentally demonstrated to exhibit $T_c$'s of $250$--$260$~K at pressures of $170$--$185$~GPa~\cite{Somayazulu2019,Drozdov2019,Hong2020}.

A very recent experimental study has reported superconductivity in the C-S-H system at pressures of $140$--$275$~GPa,
with the highest $T_c$ of $288$~K at $267$~GPa demonstrating room-temperature superconductivity~\cite{Snider2020}.
Since the underlying crystal structure has yet to be determined,  pressure-induced structural changes over the measured range cannot be ruled out.
However, the current experimental data are consistent with a continuous increase in $T_c$ with pressure
but perhaps a discontinuous $\mathrm{d}{T_c}/\mathrm{d}P$ near $230$~GPa,
suggesting a gradual structural deformation instead of an abrupt structural transition near that pressure.
More experimental data are obviously required to establish the trend of $T_c$ versus pressure, and more importantly
chemical analysis and X-ray diffraction are required to reveal the composition and crystal structure of the superconducting component in the C-S-H system.

Interestingly, the room-temperature superconductivity in the C-S-H system and its $T_c$ pressure dependence beginning below $200$ GPa~\cite{Snider2020}
are strongly reminiscent of the predicted pressure effect on $T_c$ of hole-doped H$_3$S$_{1-x}$P$_{x}$ and H$_3$S$_{1-x}$Si$_{x}$~\cite{Ge2016}.
{\color{black}Here we systematically study the effects of C and Si substitution for S on the superconductivity of H$_3$S with the cubic $Im\bar{3}m$ structure at megabar pressures}
based on first-principles calculations with the virtual crystal approximation (VCA)~\cite{Ge2016}.
{\color{black} The cubic structure is examined here since it gives the highest $T_c$ for H$_3$S~\cite{Einaga2016}; the effects of substitution on phases derived from
lower symmetry structures~\cite{Li2014,Goncharov2016,Akashi2016,Majumdar2017,Goncharov2017,Cui2020,Sun2020,Laniel} will be studied in future work.}
As demonstrated previously~\cite{Ge2016} and below,
low-level substitution can fine-tune the Fermi energy to reach the electronic density-of-states (DOS) peak and thus maximize the electron-phonon coupling.
It turns out that this can boost the $T_c$ from the original 203~K to 289~K and 283~K, respectively,
for H$_3$S$_{0.962}$C$_{0.038}$ at 260~GPa and H$_3$S$_{0.960}$Si$_{0.040}$ at 230~GPa.
The former may provide an explanation for the recent experimental observation of room-temperature superconductivity
in the C-S-H system including its overall $T_c$ pressure dependence~\cite{Snider2020}.
{The results thus also suggest an effect of C incorporation below $200$~GPa.}

\begin{figure*}[t!]
\centerline{\includegraphics[width=0.9\textwidth]{fig1.pdf}}
\caption{(a) Band structures and (b) DOS of H$_3$S at $220$~GPa (black), H$_3$S$_{0.960}$C$_{0.040}$ at $220$~GPa (solid red),
and H$_3$S$_{0.960}$C$_{0.040}$ at $240$ GPa (dashed red), with their Fermi energies set to be zero.
(c) Phonon dispersion and Eliashberg function $\alpha^2{F}(\omega)$ of
H$_3$S$_{0.960}$C$_{0.040}$ at $250$ GPa (red) and 280 GPa (blue). The magnitudes of phonon linewidths are indicated by the line thickness.
Inset: the low-frequency (magenta) and high-frequency (green) H-S bond-bending modes around $600$ $(2000)$~cm$^{-1}$
and the H-S bond-stretching modes around $2500$~cm$^{-1}$ (cyan).}
\label{fig1}
\end{figure*}

Following the approach introduced in Ref.~\onlinecite{Ge2016},
our calculations were performed within the framework of ABINIT~\cite{Gonze19971,Gonze19972,Gonze2005,Gonze2009}
using the local-density approximation.
The ion and electron interactions were treated with Hartwigsen-Goedecker-Hutter pseudopotentials~\cite{Hartwigsen1998}.
The electronic ground-state properties were calculated on a $32\times32\times32$ Monkhorst-Pack $k$-mesh using the kinetic energy cutoff of $800$~eV.
The phonon dispersions and the electron-phonon couplings were calculated on an $8\times8\times8$ $q$-grid
using the density functional perturbation theory~\cite{Baroni2001}. The atomic substitution was simulated by the self-consistent VCA,
where the virtual pseudopotentials of S$_{1-x}$Z$_x$ were set to be $V_{\rm VCA} = (1-x)V_{\rm S}+xV_{\rm Z}$.

The superconductivity of H$_3$S can be accurately described by the BCS theory, which underlines the aforementioned
strong electron-phonon coupling and high phonon frequencies as the two most important factors in producing the high $T_c$
in this class of materials.
Here we focus on the former factor, or equivalently the effect of the electronic DOS, for reasons that becomes clear below.
{\color{black}H$_3$S has a DOS that reaches $7.43$~Hartree$^{-1}$/spin at its Fermi level~\cite{Papaconstantopoulos2015},}
because of the presence of a van Hove singularity in the vicinity,
as shown in Fig.~\ref{fig1}. By substituting the S atoms with C and Si,
the H$_3$S system can be hole doped, and the Fermi level can be moved closer to the DOS peak,
and by increasing the pressure, the DOS peak can be further enhanced. Both effects are illustrated in Fig.~\ref{fig1}(b).
The dynamical stability of the crystal structure limits the substitution level to $x\leq 0.050$ for H$_3$S$_{1-x}$Z$_x$ (Z=C, Si) hereafter.

Figure~\ref{fig1}(c) compares the phonon dispersions of H$_3$S$_{0.960}$C$_{0.040}$ at $250$ and $280$~GPa.
The coupling of Fermi-level electrons with specific phonons is indicated by the thickness of the dispersion curves,
i.e., the magnitudes of phonon linewidths
\begin{equation}
\gamma_{\bm q\nu}=2 \pi \omega_{\bm q\nu}{\sum_{\substack{ij\bm{k}}}}{\vert{M^{\nu}_{i\bm k, j\bm{k+q}}}\vert}^2
\delta(\epsilon_{i\bm k}-\epsilon_{F})\delta(\epsilon_{j\bm{k+q}}-\epsilon_{F})\,,
\end{equation}
where $M^{\nu}_{i\bm k, j\bm{k+q}}$ are the microscopic electron-phonon matrix elements.
High-frequency H-S (or H-Z) bond-stretching modes near $2500$ cm$^{-1}$ have the largest phonon linewidths,
indicating strong electron-phonon coupling of the H-based vibrations.
One such high-frequency phonon mode and two lower-frequency modes are illustrated in the inset of Fig.~\ref{fig1}(c).

\begin{figure*}[t!]
\centerline{\includegraphics[width=0.9\textwidth]{fig2.pdf}}
\caption{(a) DOS (black) and electron-phonon coupling $\lambda$ (blue) of cubic H$_3$S$_{1-x}$Z$_x$ (Z=C, Si) versus $x$ at $250$~GPa.
{\color{black}Solid and dashed lines denote the results for H$_3$S$_{1-x}$C$_x$ (square) and H$_3$S$_{1-x}$Si$_x$ (circle), respectively.}
(b) DOS (black), $\lambda$ (blue), and $T_c$ (shaded) versus pressure at $x=0.040$.
For each $T_c$ curve, the upper and lower bounds are obtained by choosing $\mu^\ast=0.10$ and $0.15$, respectively.
(c) The $T_c$ maps of H$_3$S$_{1-x}$C$_x$ and H$_3$S$_{1-x}$Si$_x$ versus substitution level and pressure for $\mu^\ast=0.10$.
{\color{black}At high pressures or high substitution levels, cubic H$_3$S$_{1-x}$Si$_x$ becomes unstable; thus no data are shown.}}
\label{fig2}
\end{figure*}

The superconductivity of H$_3$S$_{1-x}$Z$_{x}$ (Z=C, Si) can be estimated by Eliashberg theory~\cite{Allen1983},
which takes into account the renormalization of electron-electron repulsion by electron-phonon interactions.
This celebrated theory has been successfully used in predictions of superconductivity in hydrogen-rich materials
as well as ambient-pressure conventional superconductors.
Figure~\ref{fig2}(a) shows that for both C- and Si-substitutions at $250$~GPa
the DOS at the Fermi level increases with increasing the substitution level $x$, reaches a maximum around $x = 0.040$,
and then starts to decrease. This trend is similar to that found for H$_3$S$_{1-x}$P$_x$~\cite{Ge2016},
and as anticipated C- or Si-substitution is almost twice as efficient as P-substitution for hole doping.
For both cases, the electron-phonon coupling $\lambda$ follows the trend of the DOS,
as the influence of low substitution on the phonon frequency is weak and secondary.
{\color{black}The upper-pressure limit of dynamical stability of cubic $Im\bar{3}m$ H$_3$S$_{1-x}$Z$_{x}$ varies with both Z and $x$.
For example, the upper-pressure limit of H$_3$S$_{0.960}$C$_{0.040}$ is 280 GPa, and that of H$_3$S$_{0.960}$Si$_{0.040}$ is 250 GPa.
Beyond these limits, acoustic phonons at the H point become imaginary signaling structural instabilities.
As seen in Fig.~\ref{fig2}(c), our study focuses on pressures of $200$--$280$~GPa and the substitution levels up to $x=0.050$.
At these pressures H$_3$S$_{1-x}$C$_{x}$ is found to be dynamically stable,
whereas H$_3$S$_{1-x}$Si$_{x}$ becomes unstable beyond about $250$ GPa with a moderate $x$-dependence.}

As shown in Fig.~\ref{fig2}(b), for both C- and Si-substitutions at $x=0.040$,
the DOS at the Fermi level increases first and then decreases as the pressure increases.
The maximum DOS occurs at different pressures for the two cases, i.e., $250$~GPa for H$_3$S$_{0.960}$C$_{0.040}$ and $230$~GPa for H$_3$S$_{0.960}$Si$_{0.040}$.
Notably, for C-substitution the DOS increases more markedly with pressure in the $230$--$250$~GPa range than that at lower pressures.
A possible reason is the complex influence of compression on the electronic structure around the Fermi level,
as implied in Fig.~\ref{fig1}(b) by the different DOS behavior at different pressures for a fixed doping level.
By contrast, the $\lambda$'s in both cases increase monotonically with pressure.
The different DOS and $\lambda$ trends with pressure are likely due to softening of the phonons around $600$~cm$^{-1}$.
As marked by the magenta arrows in the inset of Fig.~\ref{fig1}(c),
these softened phonon modes, which also contribute to $\lambda$, are low-frequency H-S bond-bending modes~\cite{Errea2015}.
As we see from $\lambda = 2\int\omega^{-1}\alpha^2\rm{F}(\omega)\rm{d}\omega$, the phonon softening is beneficial for enhancing $\lambda$.
Indeed, there is a drop in the logarithmically averaged phonon frequency ${\langle\omega\rangle}_{\log}$ from $1364$~K at $250$~GPa to $1269$~K at $280$~GPa.
This explains why $\lambda$ exhibits a modest upward trend while the DOS decreases with increasing pressure.

Given the averaged phonon frequency ${\langle\omega\rangle}_{\log}$, effective Coulomb repulsion $\mu^\ast$, and electron-phonon coupling $\lambda$,
the Allen-Dynes-modified McMillan formula~\cite{Allen1975,Durajski}
\begin{eqnarray}
	T_c&=f_1f_2&\frac{\langle\omega\rangle_{\log}}{1.20}
	\exp{\left[-\frac{1.04(1+\lambda)}{\lambda-\mu^\ast(1+0.62\lambda)}\right]}
	\label{eq:tc}
\end{eqnarray}
can be implemented to predict the $T_c$. Here $f_1$ and $f_2$ are the strong coupling and shape correction factors~\cite{Allen1975}, respectively;
a reasonable range of $\mu^\ast$ is between $0.10$ and $0.15$~\cite{range}.
For H$_3$S at $200$~GPa~\cite{Drozdov2015}, our calculations yield the $T_c$ of $194$~K for $\mu^\ast=0.12$ and $203$~K for $\mu^\ast=0.11$.
Following the behavior of the DOS, the $T_c$ increases faster in the $230$--$250$~GPa range than that at lower pressures,
as shown in Fig.~\ref{fig2}(b).
This behavior appears to parellel the upturn in $T_c$ above $230$~GPa observed experimentally in the C-S-H system~\cite{Snider2020}.
While the DOS drops with increase in pressure as discussed above,
the phonon softening enhances $\lambda$ but weakens ${\langle\omega\rangle}_{\log}$.
This implies a decrease of Debye temperature in the BCS theory.
This suggests a maximum $T_c$ versus pressure that tracks the behavior of the DOS.
In order to display the joint influence of hole doping and high pressure, as well as to identify the maximum $T_c$,
we plot the map of the $T_c$ versus the substitution level and pressure for $\mu^\ast=0.10$ in Fig.~\ref{fig2}(c).
In particular, the highest $T_c$ are $289$~K for H$_3$S$_{0.962}$C$_{0.038}$ at $260$~GPa and
$283$~K for H$_3$S$_{0.960}$Si$_{0.040}$ at $230$~GPa.
The former is very close to the highest $T_c$ of $288$~K at $267$~GPa observed in the C-S-H experiment~\cite{Snider2020}.
{\color{black}Evidently in Fig.~\ref{fig2}(c), the highest $T_c$'s are reached near the structural instabilities. This appears to be consistent with the picture that
soft phonon modes can be important for $T_c$ enhancement near structural instabilities
in superconducting hydrides~\cite{Cohen1972,Quan2019,Chen2020}.}

In conclusion, we have examined the effects of hole doping on the superconductivity of H$_3$S
with the $Im\bar{3}m$ structure at megabar pressures by using the first-principles calculations with the VCA.
This fine-tunes the Fermi energy to reach the peak in the electronic DOS, maximizes the electron-phonon coupling,
and boosts the $T_c$ to 289~K and 283~K, respectively, for H$_3$S$_{0.962}$C$_{0.038}$ at 260~GPa and H$_3$S$_{0.960}$Si$_{0.040}$ at 230~GPa.
Because of the fewer valence electrons and the lighter atomic masses,
the C- and Si-substitutions are more efficient in raising $T_c$ than substitution by P~\cite{Ge2016}.
Although less stable at the higher pressure, Si-substitution raises the $T_c$ more than substitution by C below $240$~GPa.
Most importantly, the C-substitution may provide an explanation for the recent experimental observation of room-temperature superconductivity
in the C-S-H system and its $T_c$ pressure dependence above $200$~GPa~\cite{Snider2020}.
Our findings indicate that hole doping in general--not limited to C-, Si-, and P-substitutions--is a robust approach to maximize the $T_c$ of H$_3$S.
Looking forward, our study, together with Ref.~\onlinecite{Ge2016},
opens a new avenue for substantially raising the already high $T_c$'s of hydrogen-rich materials
and calls for experimental investigation to systematically optimize the doping of these materials under pressure to reach still higher $T_c$'s.

\begin{acknowledgments}
We are grateful to Roald Hoffmann and Eva Zurek for valuable comments on this work.
F.Z. is grateful to Anvar Zakhidov, Bing Lv, and Mikhail Eremets for valuable discussions at the initial stage of this work.
The work is supported by the National Key R\&D Program of China (Grant No. 2020YFA0308800),
the National Natural Science Foundation of China (Grants Nos. 11904312 and 11734003),
the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB30000000),
the Project of Hebei Educational Department (Grants No. QN2018012), the UT Dallas Research Enhancement Fund,
the US National Science Foundation (Grant Nos. DMR-1933622 and DMR-1809649),
and the US Department of Energy (Grant Nos. DE-SC0020340 and DE-NA0003975).
\end{acknowledgments}