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On Noncompact Minimal Sets of the Geodesic Flow

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Abstract

We study nontrivial (i.e., containing more than one orbit) minimal sets of the geodesic flow on Γ\T 1ℍ2, where Γ is a nonelementary Fuchsian group. It is not difficult to prove that nontrivial compact minimal sets always exist. We establish the existence of nontrivial noncompact minimal sets in two cases: (1) Γ is a Schottky group of special kind generated by infinitely many hyperbolic elements, (2) Γ contains a parabolic element (in particular, Γ = PSL(2, ℤ)). This is done by geometric coding of geodesic orbits and constructing a minimal set for symbolic dynamics with infinite alphabet.

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Authors and Affiliations

  1. Institut de Mathematiques de Rennes 1, Campus de Beaulieu 35042, Rennes Cedex, France

    F. Dal'Bo

  2. All-Russian Institute of Electrotechnics, 143500, Istra, Moscow region, Russia

    A. N. Starkov

  3. Department of Mechanics and Mathematics, Moscow State University, 117234, Moscow, Russia

    A. N. Starkov

Authors
  1. F. Dal'Bo
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  2. A. N. Starkov
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Dal'Bo, F., Starkov, A.N. On Noncompact Minimal Sets of the Geodesic Flow. Journal of Dynamical and Control Systems 8, 47–64 (2002). https://doi.org/10.1023/A:1013900700506

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  • DOI: https://doi.org/10.1023/A:1013900700506

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