\section{introduction}

A finite area hyperbolic 2-orbifold $S$ is the quotient of the Poincar\'e upper-half plane $\HH$ by a discrete group $\Gamma\subset \text{PSL}_2(\RR).$ Let $\pi:\HH\rightarrow
S=\HH/\Gamma$ denote the covering projection and let $q\in S$ be a non-cone point of $S$. Without loss of generality, suppose that $i\in \HH$ is a preimage of $q$. As usual, $B_r(x)$ is the open disc of radius $r$ centered at $x$.  There is a number $R>0$  for which the map $\pi: B_R(i)\rightarrow B_R(q)$ is an embedding.

Each vector $v$ in the unit tangent bundle $T_1S$ uniquely determines a geodesic ray $\gamma=\gamma(v)$ which is the projection of the orbit of the geodesic flow, $\{G_t(v)\,|\,t\in [0,\infty)\},$ to $S$. If the ray keeps returning to the disc $B_r(q)$, then there is a sequence of pairs of parameter values $\{(t_i,s_i)\}$ so that 
\begin{equation*}
\bigcup_{i=1}^{\infty}\gamma([t_i,s_i])=\gamma\cap \overline{B}_r(q),
\end{equation*}
in other words, $\gamma$ meets the closed disc $B_r(q)$ in precisely the arcs $\gamma([t_i,s_i])$ of $\gamma.$ We shall refer to these arcs as the excursions of $\gamma$ into $B_r(q).$
For $v\in T_1S$ define the counting function $N_v(r)(t)=\max\{i\,|\,t_i\leq t\}.$ Then we have

\begin{thm}\label{return0}
For almost all $v\in T_1S$ and all $0<r<R$
\begin{equation}\label{average}
\lim_{t\rightarrow\infty}\frac{1}{t}N_v(r)(t)=\frac{2\sinh r}{ area (S)}.
\end{equation}
\end{thm}

Two  consequences of the theorem are given in

\begin{cor}\label{return1}
For almost all $v\in T_1S$ and all $0<r<R$
\begin{itemize}

\item[{\rm (i)}] The asymptotic average  length of an excursion is
\begin{equation}\label{part1}
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n} (s_i-t_i)=\frac{\pi(\cosh r-1)}{\sinh r}.
\end{equation}

\item[{\rm (ii)}]
The average length of an arc joining the beginnings of consecutive excursions is 
\begin{equation}\label{part2}
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n-1} (t_{i+1}-t_{i})=\frac{ area (S)}{2\sinh r}.
\end{equation}
 

\end{itemize}
\end{cor}

Note that when $r$ is large, (\ref{part1})   is close to $\pi.$ 
 In Section \ref{3} we expand these results to cover the case where $q$ is a cone point. We will also  consider their expressions  in terms of the  area of a disc rather than  its radius. 
 

 \section{Details}

The unit tangent bundle of $\HH$ is $T_1\HH=\HH\times S^1.$ In these coordinates the geodesic flow $\tilde{G}$ has the natural invariant measure $\tilde{m}=dAd\theta.$ The non-vertical vectors in $T_1\HH$ are a set of full measure and can be parameterized by the set of triples $(x,y,t)\in\RR^3$ with $x\not=y.$ Let $\alpha$ be the geodesic in $\HH$ oriented from the  endpoint $\alpha_{-}=y$ to $\alpha_{+}=x $  and parameterized so that $\alpha(0)$ is the Euclidean midpoint of the semicircle $\alpha(\RR)$. Then $(x,y,t)$ corresponds to the vector $v=\dot{\alpha}(t)\in T_1\HH.$ In these coordinates the geodesic flow has the form $\tilde{G}_s(x,y,t)=(x,y,t+s).$ Furthermore, the geodesic flow on $T_1S$ has the invariant probability measure $\mu$, whose lift to $T_1\HH$ is equal to
\begin{equation}
\tilde{\mu}= \frac{1}{2\pi\,\text{area}(S)}\tilde{m}=\frac{1}{\pi\, \text{area}(S)}\varphi(x,y)\, dx dy dt.
\end{equation}
where $\varphi (x,y)=(x-y)^{-2}$ \cite{nicholls}.  

Let $\mathcal{E}\subset T_1S$ be the set of vectors $v$ for which there are infinitely many excursions of $\gamma=\gamma(v)$ into $B_r(q)$ for each $0<r<R.$ As a consequence of  the Poincar\'e recurrence Theorem,  \cite{sinai}, $\mathcal{E}$ has full measure in $T_1S$. Observe that if $v\in \mathcal{E},$ then $G_t(v)\in \mathcal{E}$ for all $t\in \RR.$

Let $\mathcal{L}_r^*$ be the subset of the unit tangent bundle over the circle $\partial B_r(q)$,  (bounding the disc $B_r(q)$)  consisting of vectors that point into the disc $B_r(q).$  Then $\mathcal{L}_r=\mathcal{L}_r^*\cap\mathcal{E}$ is a cross-section for the geodesic flow on $T_1S$, \cite{bks}. In other words, for almost all $v\in T_1S$ there exists an increasing sequence of values $\tau_i$ so that $G_{\tau_i}(v)\in \mathcal{L}_r$.
Given $\epsilon<R-r$, define the $\epsilon$-thickened section $ \mathcal{L}_r(\epsilon)=\{G_t(v)\,|\, t\in [0,\epsilon], v\in \mathcal{L}_r\}.$ Analysis of the thickened section is the main tool in the proof of Theorem \ref{return0}. This is similar to the approaches taken in \cite{haas}, \cite{ nakada} and \cite{strat}. 

\begin{prop}\label{thick}
\begin{equation}
\mu( \mathcal{L}_r(\epsilon))=\frac{2\epsilon\sinh r}{ area (S)}
\end{equation}
\end{prop}

The cross-section $\mathcal{L}_r$ lifts to a subset $\tilde{\mathcal{L}}_r$ of $T_1\HH$ over the circle $\partial B_r(i)$ in $\HH$. Then $ \mathcal{L}_r(\epsilon)$ has a connected preimage $\Lambda_r(\epsilon)\supset \tilde{\mathcal{L}}_r(\epsilon)$ in $T_1\HH.$ Furthermore, since $\pi$ is an embedding of $B_r(q)$ into $S$ and $\epsilon<R-r$, the projection $\pi_{*}:T_1\HH\rightarrow T_1S$ restricts to an embedding  of $\Lambda_r(\epsilon)$ onto $ \mathcal{L}_r(\epsilon)$. It follows that $\Lambda_r(\epsilon)=  \tilde{\mathcal{L}}_r(\epsilon).$ Thus $\mu( \mathcal{L}_r(\epsilon))=\tilde{\mu}( \tilde{\mathcal{L}}_r(\epsilon))$ and we can perform computations in $\RR^3$ using the coordinates given earlier.  

 
Define the M\"obius transformations:
\begin{equation*}
W_{\rho}(x)=\displaystyle{\frac{z\sinh\rho -1}{z+\sinh\rho}}\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,  
U_{\rho}(x)=\displaystyle{\frac{z\sinh\rho +1}{-z+\sinh\rho}}.
\end{equation*}

For real numbers $a\not= b$ let $\overline{ab}$ denote the geodesic $\alpha$ in $\HH$ with endpoints $\alpha_+=a$ and $\alpha_-=b.$


We will need the following extension of Theorem 5 from \cite{haas3}.

\begin{thm}\label{coneold}
Given $x\geq 0$, the geodesics $\overline{xw}$ and $\overline{xu}$ with $w=W_{\rho}(x)\in [-1/x,x)$ and $u=U_{\rho}(x)$, are both tangent to the disc $B_{\rho}(i)$. Furthermore, given $x\leq 0$, the geodesics $\overline{xw}$ and $\overline{xu}$ with $w=-W_{\rho}(-x)\in (x,-1/x]$ and $u=-U_{\rho}(-x)$, are both tangent to the disc $B_{\rho}(i).$  

\end{thm}

Although the proof in \cite{haas3} only addressed the tangent geodesic with endpoints $x$ and $W_{\rho}(x)$, the other case can be uncovered there and follows by choosing $\xi=z(c+i\sqrt{1-c^2})$ in Lemma 3 and $\eta=c+i\sqrt{1-c^2}$ in the proof of Theorem 5.
\vskip .1in
  
\noindent {\em Proof of Proposition \ref{thick}}.\,
Fix $0<r<R$. For a given $x\in \hat{\RR}$, let $I_x$ denote the set of values $y\in \hat{\RR}$ for which the geodesic $\overline{xy}$ intersects the closed disc $\overline{B}_r(i)$. It follows from Theorem \ref{coneold} that for $x\geq 0,\, I_x$ is the interval between the points $W_{\rho}(x)$   and $U_{\rho}(x)$; the unbounded interval if $x \leq \sinh \frac{r}{2}$ and the bounded interval if  $x>\sinh  \frac{r}{2}$. A similar statement is true for negative $x$.

For $x\in \RR$ and $y\in I_x$, let $t_{xy}$ denote the parameter value for which the unit tangent vector   $(x,y,t_{xy})\in \tilde{\mathcal{L}}_r.$
Then
\begin{equation*}
 \tilde{\mathcal{L}}_r(\epsilon) =\{ (x,y,t)\,|\,x\in\RR,\, y\in I_x\, \text{and}\, t_{xy}\leq t\leq t_{xy}+\epsilon\}.
\end{equation*}
Consequently, $ \tilde{\mu}(\tilde{\mathcal{L}}_r(\epsilon))=$
\begin{equation}\label{area}
\frac{1}{\pi\,\text{area}(S)}\int_{\RR}\int_{I_x}\int_{t_{xy}}^{t_{xy}+\epsilon}\varphi(x,y)\,dtdydx = \frac{\epsilon}{\pi\,\text{area}(S)}\int_{-\infty}^{\infty}\int_{I_x}\varphi(x,y)\,dydx.  
\end{equation}
This unpacks as the following sum of integrals:
\begin{equation*} 
\frac{\epsilon}{\pi\,\text{area}(S)}\left [\int_0^{\sinh  \frac{r}{2}}\int_{-\infty}^{W_r(x)}\frac{1}{(x-y)^2}\,dydx+
\int_0^{\sinh  \frac{r}{2}}\int_{U_r(x)}^{\infty}\frac{1}{(x-y)^2}\,dydx+\right .
\end{equation*}

\begin{equation*}
\hskip .8in \int_{\sinh  \frac{r}{2}}^{\infty}\int_{U_r(x)}^{W_r(x)}\frac{1}{(x-y)^2}\,dydx+
\int_{-\sinh  \frac{r}{2}}^0\int_{-W_r(-x)}^{\infty}\frac{1}{(x-y)^2}\,dydx+
\end{equation*}
\begin{equation*}
\hskip 1in\left . \int_{-\sinh  \frac{r}{2}}^0\int_{-\infty}^{-U_r(-x)}\frac{1}{(x-y)^2}\,dydx+
\int_{-\infty}^{-\sinh  \frac{r}{2}}\int_{-W_r(-x)}^{-U_r(-x)}\frac{1}{(x-y)^2}\,dydx \right ]
\end{equation*}
The  integrals are easily computed.\\  
\hfill $\Box$

\noindent {\em Proof of Theorem \ref{return0}}.\,
 Let $\chi_Y(t)$ denote the characteristic function of the set $Y$. Then for $\epsilon<R-r$ we have the inequalities
 \begin{equation*}
 \int_0^t\chi_{\mathcal{L}_r(\epsilon)}(G_{\tau}(v))d\tau\,-2\epsilon\,\leq\, \epsilon N_v(r)(t)\,\leq\,
  \int_0^t\chi_{\mathcal{L}_r(\epsilon)}(G_{\tau}(v))d\tau\,+2\epsilon.
  \end{equation*}

Divide thru by $\epsilon t$. Since  the geodesic flow is ergodic, as $t\rightarrow\infty$ the left and right hand limits approach 
$(1/\epsilon)\mu(\mathcal{L}_r(\epsilon)).$
The theorem then follows from Proposition \ref{thick}.
\hfill $\Box$

\vskip .1in

\noindent {\em Proof of Corollary \ref{return1}}.\,
First we  address part (ii). Let $v\in\mathcal{E}$ and let $\{(t_i,s_i)\}$ be the sequence of excursions into the disc $B_r(q)$ associated to the geodesic $\gamma=\gamma(v).$ In particular we have $N_v(r)(t_n)=n$ (or $n-1$ if $\gamma(0)\in B_r(q))$. Then by Theorem \ref{return0},  for almost all $v\in T_1S$
\begin{equation*}\label{coravg}
\lim_{n\rightarrow\infty}
\frac{1}{n}\sum_{i=1}^{n-1}(t_{i+1}-t_i)=
\lim_{n\rightarrow\infty}\frac{t_n}{n}=
\lim_{n\rightarrow\infty}\frac{t_n}{N_v(r)(t_n)}
=\frac{\text{area}(S)}{2\sinh r},
\end{equation*}
 proving part (ii).

For part (i) observe that the limit (\ref{part1}) can be expressed as an integral:
\begin{equation}\label{flow}
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n} (s_i-t_i)=\lim_{n\rightarrow\infty}\frac{1}{n}\int_0^{s_n}\chi_{(B_r(q))}\left(G_v(\tau)\right)d\tau
\end{equation}
By Theorem \ref{return0} and the Ergodic Theorem for flows, this can be further divided into a product of limits, each piece of which converges for almost all $v\in T_1S$. Thus (\ref{flow}) is equal to 
\begin{equation}\label{split}
\begin{split}
& \lim_{n\rightarrow\infty}\frac{1}{s_n}\int_0^{s_n}\chi_{(B_r(q))}\left(G_v(\tau)\right)d\tau\,\times\, 
\lim_{n\rightarrow\infty}\frac{s_n}{n}\\
\\
& =\frac{ \text{area}(B_r(q))}{\text{area}(S)}\,\times\,\frac{\text{area}\,(S)}{2\sinh r}=\frac{\pi(\cosh r-1)}{\sinh r}
\end{split}
\end{equation}
as asserted in part (i) of the Corollary, \cite{Beardon}. 
\hfill $\Box$


\section{Cone points}\label{3}
 
If the stabilizer of $i$ in $\Gamma$ is a subgroup of order $k>0$ then $q$ is called a cone point of order $k$. Let $R>0$ be the largest value so that for all $0<r<R$ the projection  $\pi: B_r(i)\rightarrow B_r(q)$ is precisely $k-\text{to}-1$ in the complement of $i$. Then for $0<r<R$, the definition of $N_v(r)(t)$ extends without modification to the case where $q$ is a cone point. The thickened sectioned is defined as before but now,  since  the map $\pi_*$ is $k-\text{to}-1$, instead of Proposition \ref{thick}, we have
\begin{equation}\label{withcone}
\frac{1}{\epsilon}\mu( \mathcal{L}_r(\epsilon)= \frac{1}{k\epsilon}\tilde{\mu}( \tilde{\mathcal{L}}_r(\epsilon))=\frac{2\sinh r}{k\,\text{area}(S)}.
\end{equation}
Thus Theorem \ref{return0} remains valid when $q$ is a cone point of order $k$, with the right hand side of (\ref{average}) replaced by (\ref{withcone}).

 
  
When  studying the rate of geodesic return to the neighborhood of a cone point, it is  interesting to have all the quantities   expressed  in terms of area rather than radius. Let $D_a(q)$ denote the disc of area $a$ about $q$. 
Observe that, if $q$ is a cone point of order $k$, then the area of the disc $B_r(q)$  is 
\begin{equation*}
\text{area}(B_r(q))=\frac{1}{k}\text{area}(B_r(i))= \frac{2\pi}{k} (\cosh r -1). 
\end{equation*}
Thus $D_a(q)$ is a disc or radius $r=\cosh^{-1}(\frac{k}{2\pi}a+1).$ 

Let  $\{(t_i,s_i)\}$ be the sequence of excursions of the geodesic $\gamma=\gamma(v)$ into the disc $D_a(q) $. Then, almost as before, for $v\in\mathcal{E}$ we can define $\mathcal{N}_v(a)(t)=\max\{i\,|\,t_i\leq t\}.$ In this setting  Theorem \ref{return0} for a cone point of order $k$,  expressed in terms of area, takes the form:
\begin{thm}\label{area}
For almost all $v\in T_1S$ and all $0<a<A=(2\pi/k) (\cosh R -1)$
\begin{equation}\label{average2}
\lim_{t\rightarrow\infty}\frac{1}{t}\mathcal{N}_v(a)(t)=\frac{2\sqrt{(\frac{1}{k}+\frac{1}{2\pi}a)^2+\frac{1}{k^2}}}{\text{area}\,(S)}.
\end{equation}
\end{thm}
When $k$ is large this value is close to  $a/(\pi\,\text{area}(S))$, which  it the value computed in \cite{haas} for the asymptotic rate of return to a cusp neighborhood.


Also,   the asymptotic average length of an excursion can still be computed when $q$ is a cone point. Replacing $B_r(q)$ by $D_a(q)$, equation (\ref{flow}) holds by definition and, as in equation (\ref{split}), using Theorem \ref{area} and  the Ergodic Theorem we have
\begin{cor}\label{areaavg}
For almost all $v\in T_1S$ and all $0<a<A$, the average length of an excursion into $D_a(q)$ is
\begin{equation*}
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n} (s_i-t_i)=  \frac{a}{\text{area}(S)}\times
\frac{\text{area}(S)}{2\sqrt{(\frac{1}{k}+\frac{1}{2\pi}a)^2+\frac{1}{k^2}}}=
\frac{\pi}{\sqrt{(\frac{2\pi}{ka}+1)^2+(\frac{2\pi}{ka})^2}}
\end{equation*}
\end{cor}

For $k$ large the average length of an excursion is close to $\pi$, which was the value  computed in \cite{haas} for the average length of an excursion relative to a cusp neighborhood. When the area of the disc is large, the average length of an excursion is also close to $\pi,$  independent of the order of the cone point.



