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).

- 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 $S$)?
- 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.