Ratio limits and pressure function for group extensions of Gibbs Markov maps

TL;DR

This paper extends ratio limit theorems to deterministic walks on groups driven by Gibbs Markov maps, providing new insights into pressure functions without symmetry assumptions.

Contribution

It generalizes ratio limit theorems to deterministic group walks driven by Gibbs Markov maps and characterizes pressure functions without symmetry.

Findings

Established ratio limit theorems for deterministic walks on groups

Provided a characterization of the pressure function without symmetry

Addressed the challenge of the absence of convolution structure

Abstract

Ratio limit theorems for random walks on (various) groups are known. We obtain a generalization of this type of ratio limit for deterministic walks on certain groups driven by Gibbs Markov maps. In terms of proofs, the main difficulty comes down to the absence of a convolution structure. Also, for (finitely generated) group extensions of Gibbs Markov maps we obtain a characterization of the pressure function without a symmetry assumption.