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October  2009, 3(4): 637-646. doi: 10.3934/jmd.2009.3.637

Morse coding for a Fuchsian group of finite covolume

Arseny Egorov 1,

1. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received  November 2009 Revised  December 2009 Published  January 2010

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: Fuchsian group, Morse coding of geodesics, Dirichlet domain., topological Markov chain, cutting sequence.
Mathematics Subject Classification: Primary: 37D40, 37B10; Secondary: 20H1.
Citation: Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637
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