Schottky groups and symbolic dynamics

Abstract

Throughout this chapter, the group Γ will designate a Schottky group generated by two hyperbolic isometries g ₁,g ₂ (see Sect. II.1). By definition, such a group admits a Dirichlet domain centered at a point designated to be 0 in the Poincaré disk. The possible cases are diagrammed below in Fig. IV.1. For further details, the reader may refer to Sect. II.1.

The goal of this chapter is to encode the trajectories of the geodesic flow restricted to Ω _g (T ¹ S) into doubly-infinite sequences, and to develop this point of view into a method of studying the dynamics of this flow. This symbolic approach will allow us to present new proofs of Theorems III.3.3 and III.4.2. Moreover we will complete the latter theorem by characterizing the dense trajectories of Ω _g (T ¹ S) in terms of sequences. As applications, we will construct, in the general case of a non-elementary Fuchsian group Γ′, trajectories of the geodesic flow on $\varOmega_{g} (\varGamma' \backslash T^{1}\mathbb{D})$ which are neither periodic nor dense.