Topological dynamics of the horocycle flow

Part of the Universitext book series (UTX)


In this chapter, we analyze the topology of the trajectories of another classical example of flow on the quotient of T 1ℍ by a Fuchsian group: the horocycle flow. Our method is based on a correspondence between the set of horocycles of ℍ and the space of non-zero vectors in ℝ2 modulo {±Id}. This vectorial point of view allows one to relate the topological dynamics of the linear action on ℝ2 of a discrete subgroup Γ of SL(2,ℝ) to that of the horocycle flow on the quotient of T 1ℍ by the Fuchsian group corresponding to Γ. In the geometrically finite case, we show that the horocycle flow is less topologically turbulent than the geodesic flow (Sect. 4).

Throughout this chapter, we use the definitions and notations associated with the dynamics of a flow as originally introduced in Appendix A.


Modular Group Fuchsian Group Periodic Trajectory Vectorial Point Schottky Group 


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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.IRMARUniversité Rennes 1Rennes CedexFrance

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