# Possible behaviors of geodesics on hyperbolic surface

I would like to ask about possible behaviors of geodesics on closed hyperbolic surface. I only know some kinds:

1. Closed geodesics
2. Asymptotic to closed geodesics in both directions
3. Asymptotic to a closed geodesic in one direction, dense in another direction
4. Dense in both directions.

Are there other kinds? And how about the case of non-compact hyperbolic surface? Thanks in advance

• Can you say more explicitly what you mean by "dense in a direction"? For example, do you mean that the closure of a ray in that direction is equal to the whole surface? – Lee Mosher Sep 18 at 12:15
• Yes, exactly what I mean. We can consider the limit set of the forward orbit ($\omega$-limit set). I am thinking about the modular surface when looking at the geodesic ray begin from $\sqrt[3]{2}$ – Dynamic Sep 18 at 12:17
• I meant the geodesic ray end up to $\sqrt[3]{2}$ – Dynamic Sep 18 at 12:23
• I've added a complete description of some counterexamples. – Lee Mosher Sep 19 at 0:52

The first source is the theory of geodesic laminations on surfaces (I would look this up not just on wikipedia but in the original sources by W. Thurston, and/or the book by Casson and Bleiler). In this theory are lots of new types of geodesics for which the parameterizing map --- which is $\mathbb R \mapsto S$ for a non-closed leaf and $S^1 \mapsto S$ for a closed leaf --- is one-to-one, also known as simple geodesics. If the resulting subset of $S$ is denoted $\lambda$, and if I let $\Lambda$ denote the the closure of $\lambda$, then $\Lambda$ will always have measure zero, so it is never equal to $S$ (this was known by Nielsen). Also, $\Lambda$ will have a unique decomposition as a union of simple geodesics. This kind of subset $\Lambda$ is called a geodesic lamination on $S$. One way to construct interesting geodesic laminations is by taking a sequence $$c_0,c_1,c_2,c_3,...$$ of closed geodesics whose lengths go to infinity, and then taking any limit point of this sequence in the space of closed subsets of $S$ with the Hausdorff topology. If you choose $f : S \to S$ to be a homeomorphism whose mapping class is infinite order and irreducible, and if you define $c_i$ as the closed geodesic homotopic to $f^i(c_0)$, then every leaf of the limit $\Lambda$ will be a non-closed leaf that fails to satisfy your four conditions.
Another source is the theory of Markov partitions. The geodesic flow on $T^1 S$ has a Markov partition; this was known to E. Hopf. The Markov partition is encoded by a finite directed graph $\Gamma$; the flow lines (i.e. the geodesics on $S$) correspond one-to-finite with the bi-infinite directed paths in $\Gamma$. If you now take a proper subgraph of $\Gamma$ which is strongly connected and is not just a circle, and then you take any bi-infinite directed path in that subgraph which does not cycle around a periodic closed path in either direction, then the corresponding infinite geodesic fails to satisfy your four conditions.