The geodesic flow on a compact hyperbolic surface (i.e. a surface with a riemannian metric of constant curvature $-1$) has been well-studied, in particular it has been known for a long time that it is ergodic (in fact mixing). On an hyperbolic surface with infinite volume however, I am not aware of any result about dynamical properties of this flow (there seems to be some results about infinite translation surfaces however). I welcome any reference to results in this direction as an answer (or perhaps more appropriately as a comment) to this question. 


Here are some more specific queries: let $S$ be an infinite-volume complete hyperbolic surface (for simplicity let us assume that $S$ has no cusps; if necessary one could also assume that $S$ has positive injectivity radius, though I would rather avoid making this hypothesis).


1. If $S$ has an open subset $U$ of ends which is a Cantor set (i.e. a neighbourhood of $U$ is diffeomorphic to the boundary of a regular neighbourhood of a tree embedded in $\mathbb{R}^3$), does the geodesic flow of $S$ starting at some basepoint return to any neighbourhood of an end in $U$ infinitely many times with positive probability (depending uniformly on the basepoint in a compact subset of $S$)?
2. If $E$ is an isolated end of $S$ with linear or quadratic volume growth, does the geodesic flow leave any neighbourhood of $E$ with probability one?


Note that these questions are analogues for surfaces of the corresponding facts for random walks on infinite graphs (of bounded valency). 


For convenience let me recall here the definition of an end of a surface. For this one needs to fix a sequence $K\_i$ of compact subsets, with $K\_i\subset K\_{i+1}$ and $\bigcup\_i K\_i = S$. An end of $S$ is then determined by a sequence $C\_i$ where $C\_i$ is a connected component of $S\setminus K\_i$ and $C\_{i+1}\subset C\_i$. The topology on the set of ends has for a basis of open sets the sets of ends starting with a given finite sequence $C\_1\supset\ldots \supset C\_n$, for all such sequences. 



Geodesic flow on infinite surfaces
Dynamics of the geodesic flow for infinite volume manifolds has been studied a lot, but with respect to the most relevant measure, that is the Bowen-Margulis-Patterson-Sullivan measure. A complete reference could be Roblin, Mémoires SMF. 


But I guess that you are interested only in the Lebesgue/Liouville measure ? 


In the examples you have in mind, the geodesic flow will typically be dissipative. 


First, the case of abelian covers of compact hyperbolic surfaces is well understood. The geodesic flow is ergodic in the case of Z or Z^2 covers, and totally dissipative in the other cases. This is due to Mary Rees and others before her. 


This let believe that the answer to your question 2 could be positive. But to my knowledge, nothing precise is known, for both questions. 


If the surface you are interested in is a regular cover of a compact hyperbolic surface, I guess that lifting the coding of the geodesic flow of the compact surface to the cover could allow to use the probabilistic results on random walks. (In their common work, Ledrappier-Sarig use such coding lifted to the cover.) 


However, if the surface is not a cover, I don't know any method to prove it. 


Bests


Barbara


