One-Parameter Deformations of the Bowen-Series Map Associated to Cocompact Triangle Groups

In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on the unit circle tightly associated with this group. In general, their fundamental domain enjoys what has since been called the 'extension property'. We determine the exact set of signatures for cocompact triangle groups for which this extension property can hold for any convex fundamental domain, and verify that for this restricted set, the Bowen-Series fundamental domain does have the property. To each Bowen-Series function in this corrected 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, we show that 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. Finally, we prove that there exist functions in the one-parameter deformation family that are not orbit equivalent to the action of the group on the unit circle