Extremal parameters and their duals for boundary maps associated to Fuchsian groups

Abstract

We describe arithmetic cross-sections for geodesic flow on compact surfaces of constant negative curvature using generalized Bowen–Series boundary maps, and their natural extensions, associated to co-compact torsion-free Fuchsian groups. If the boundary map parameters are extremal—that is, each is an endpoint of a geodesic that extends a side of the fundamental polygon—then the natural extension map has a domain with finite rectangular structure, and the associated arithmetic cross-section is parametrized by this set. This construction allows us to represent the geodesic flow as a special flow over a symbolic system of coding sequences. Moreover, each extremal parameter choice has a corresponding dual parameter choice such that the “past” of the arithmetic code of a geodesic is the “future” for the code using the dual parameter. This duality was observed for two classical parameter choices by Adler and Flatto; here we show constructively that every extremal parameter choice has a dual.