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

  • $\begingroup$ 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? $\endgroup$ – Lee Mosher Sep 18 '18 at 12:15
  • $\begingroup$ 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 $\sqrt[3]{2}$ $\endgroup$ – Dynamic Sep 18 '18 at 12:17
  • $\begingroup$ I meant the geodesic ray end up to $\sqrt[3]{2}$ $\endgroup$ – Dynamic Sep 18 '18 at 12:23
  • $\begingroup$ I've added a complete description of some counterexamples. $\endgroup$ – Lee Mosher Sep 19 '18 at 0:52

There are two sources for examples that go beyond the cases you listed.

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.


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