Journal of Modern Dynamics (JMD)

Morse coding for a Fuchsian group of finite covolume

Pages: 637 - 646, Issue 4, October 2009      doi:10.3934/jmd.2009.3.637

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Arseny Egorov - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)

Abstract: We consider a Fuchsian group Г and the factor surface H/Г, which has constant curvature $-1$ and maybe a few singularities. If we lift the surface continuously to $\H$ (except for a subset of a lower dimension), we obtain a fundamental domain $\D$ of Г. This can be done in different ways; ours is to restrict the choice to so-called Dirichlet domains, which always are convex polygonal subsets of $\H$. Given a generic geodesic on $\H$, one can produce a so-called geometric Morse code (or the cutting sequence) of the geodesic with respect to $\D$. We prove that the set of Morse codes of all generic geodesics on $\H$ with respect to $\D$ forms a topological Markov chain if and only if $\D$ is an ideal polygon.

Keywords:  Morse coding of geodesics, cutting sequence, topological Markov chain, Fuchsian group, Dirichlet domain.
Mathematics Subject Classification:  Primary: 37D40, 37B10; Secondary: 20H10.

Received: November 2009;      Revised: December 2009;      Available Online: January 2010.