Entropy stability and Milnor–Thurston invariants for Bowen–Series-like maps

Abstract

We define a family of discontinuous maps on the circle, called Bowen–Series-like maps, for geometric presentations of surface groups. The family has 2N parameters, where 2N is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor–Thurston theory for one-dimensional maps

This work has been possible due to the financial support of our respective institutions, the MCIU grant numbers PID2020-118281GB-C31 and PID2023-146424NB-I00, and the Generalitat de Catalunya grant number 2021 SGR 00113

Open Access funding provided thanks to the CSUC agreement with Cambridge University Press (CUP)