A group from a map and orbit equivalence

TL;DR

This paper investigates the conditions under which expanding piecewise homeomorphisms of the circle can be realized as Bowen-Series maps for Fuchsian groups, exploring the relationship between group actions and dynamical systems.

Contribution

It provides a partial characterization of when such circle maps correspond to Fuchsian group actions, linking dynamical properties to geometric group theory.

Findings

Identifies conditions for a circle map to be a Bowen-Series-type map

Establishes which groups can produce such maps

Provides partial criteria for reverse construction

Abstract

In two papers published in 1979, R. Bowen and C. Series defined a dynamical system from a Fuchsian group, acting on the hyperbolic plane $\mathbb{H}^2$. The dynamics is given by a map on $S^1$ which is, in particular, an expanding piecewise homeomorphism of the circle. In this paper we consider a reverse question: which dynamical conditions for an expanding piecewise homeomorphism of $S^1$ are sufficient for the map to be a ``Bowen-Series-type" map (see below) for some group $G$ and which groups can occur? We give a partial answer to these questions.