Symbolic Dynamics for the Geodesic Flow on Two-dimensional Hyperbolic Good Orbifolds

Abstract

We consider the geodesic flow on orbifolds of the form Γ∖H, where H is the hyperbolic plane and Γ is a discrete subgroup of \PSL(2,\R). For a huge class of such groups Γ (including some non-arithmetic groups like, e.g., Hecke triangle groups) we provide a uniform and explicit construction of cross sections for the geodesic flow such that for each cross section the associated discrete dynamical system is conjugate to a discrete dynamical system on a subset of \R×\R. There is a natural labeling of the cross section by the elements of a certain finite set L of Γ. The coding sequences of the arising symbolic dynamics can be reconstructed from the endpoints of associated geodesics. The discrete dynamical system (and the generating function for the symbolic dynamics) is of continued fraction type. In turn, each of the associated transfer operators has a particularly simple structure: it is a finite sum of a certain action of the elements of L.