CSDC Seminar: One-Parameter Deformations of the Bowen-Series map associated to cocompact triangle groups (Ayse Karatas)

Fuchsian groups are groups of isometries on hyperbolic space, and to each such group one can associate a signature. In 1979, for groups with a specific signature, Bowen and Series constructed an explicit fundamental domain, and from this a function on S1 tightly associated with this group. In general, their fundamental domain enjoys what has since been called the extension property.

In this study, we consider Bowen-Series functions associated to cocompact Fuchsian triangle groups. We determine the exact set of signatures for which the extension property can hold for any fundamental domain. To each Bowen-Series function in this setting, we naturally associate four one-parameter deformation families of circle functions. We show that each of these functions is aperiodic if and only if it is surjective; and, is finite Markov if and only if its natural parameter is a hyperbolic fixed point of the triangle group at hand. Furthermore, the topological entropy is constant on the Markov, aperiodic members of each one-parameter deformation family and is equal to that of the Bowen-Series function.

This talk is supported by an LMS Scheme 5 grant.