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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 Teichm├╝ller 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)
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[1] D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Math. Inst.90 (1969), 1–235.
[2] –, On a class of invariant sets of smooth dynamical systems. Proc. Fifth Intern. Congr. Nonlinear Oscillations, Inst. Math. Acad. Sci. Ukr. SSR, Kiev2 (1970), 39–45.
[3] A. F. Beardon, The Geometry of Discrete Groups.New York-Heidelberg-Berlin:Springer-Verlag, 1983 · Zbl 0528.30001
[4] M. Burger, Horocycle flow on geometrically finite surfaces.Duke Math. J. 61 (1990), 779–803. · Zbl 0723.58041 · doi:10.1215/S0012-7094-90-06129-0
[5] S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms.Adv. Sov. Math. 16 (1993), 91–137. · Zbl 0814.22003
[6] Y. Guivarch, Proprietes ergodiques, en mesure infinie, de certains systems dynamiques fibres.Ergod. Theor. and Dynam. Syst. 9 (1989), 433–453.
[7] E. Hopf, Statistik der geodatischen Linien in Mannigfaltigkeiten negativer Krummung.Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.-Nat. Kl. 51 (1939), 261–304.
[8] G. A. Hedlund, Fuchsian groups and transitive horocycles.Duke Math. J. 2 (1936), 530–542. · JFM 62.0392.03 · doi:10.1215/S0012-7094-36-00246-6
[9] G. A. Margulis, Compactness of minimal closed invariant sets of actions of unipotent groups.Geom. Dedicata 37 (1991), 1–7. · Zbl 0733.22005 · doi:10.1007/BF00150402
[10] P. J. Nicholls, The ergodic theory of discrete groups.London Math. Soc. Notes 143 (1989). · Zbl 0674.58001
[11] –, Transitivity properties of Fuchsian groups.Can. J. Math. 28 (1976), 805–814. · Zbl 0336.30006 · doi:10.4153/CJM-1976-077-8
[12] –, Granett points for Fuchsian groups.Bull. London Math. Soc. 12 (1980), 216–218. · Zbl 0424.30039 · doi:10.1112/blms/12.3.216
[13] –, Fundamental domains of Fuchsian groups.Math. Z. 174 (1980), 187–196. · Zbl 0439.30030 · doi:10.1007/BF01293537
[14] S. J. Patterson, Some examples of Fuchsian groups.Proc. London Math. Soc. 39 (1979), 276–298. · Zbl 0411.30034 · doi:10.1112/plms/s3-39.2.276
[15] Ch. Pommerenke, On the Green’s function of Fuchsian groups.Ann. Acad. Sci. Fenn. 2 (1976), 409–427. · Zbl 0363.30029
[16] –, On Fuchsian groups of divergent type.Mich. Math. J. 28 (1981), 297–310. · Zbl 0468.30038 · doi:10.1307/mmj/1029002560
[17] –, On Fuchsian groups of accessible type.Ann. Acad. Sci. Fenn. 7 (1982), 249–258. · Zbl 0471.30036
[18] M. Rees, Checking ergodicity of some geodesic flows with infinite Gibbs measure.Ergod. Theor. and Dynam. Syst. 1 (1981), 107–133. · Zbl 0469.58012
[19] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions.Proc. Stony Brook Conf. on Kleinian groups and Riemann Surfaces (1978), 465–496.
[20] H. Shirakawa, An example of infinite measure preserving geodesic flows on a surface with constant negative curvature.Comm. Math. Univ. St. Pauli 31 (1982), 163–182. · Zbl 0506.58027
[21] M. Taniguchi, Examples of discrete groups of hyperbolic motions conservative but not ergodic at infinity.Ergod. Theor. and Dynam. Syst. 8 (1988), 633–636. · Zbl 0643.22006
[22] P. Tukia, Rigidity theorem for Mobius groups.Invent. Math. 97 (1989), 405–431. · Zbl 0674.30038 · doi:10.1007/BF01389048
[23] E. B. Vinberg and O. V. Shvartsman, Riemann surfaces. In:Itogi Nauki i Tehniki, VINITI, Algebra-Topology-Geometry, Moscow 16 (1978), 191–245.
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