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Fuchsian groups from the dynamical viewpoint. (English) Zbl 0986.37033
This paper provides an elementary introduction to Fuchsian groups theory for specialists on dynamical systems. The author paid special attention to the structure of orbits of the horocycle flow on $$T^1M$$ of the corresponding hyperbolic surface $$M=\Gamma\setminus\Delta$$, where $$\Gamma$$ is the Fuchsian group. He constructs Fuchsian groups with new types of horocycle orbits which are neither closed nor dense in the nonwandering set. Moreover, the author presents both a unique classification of Fuchsian groups from the dynamical point of view and some open problems.

##### MSC:
 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
##### Keywords:
Fuchsian group; horocycle flow; nonwandering set
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##### References:
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