Generating Functions for Special Flows over the 1-Step Countable Topological Markov Chains

TL;DR

This paper develops generating functions for special flows over countable topological Markov chains, providing formulas for entropy calculation and conditions for maximal entropy measures, with applications to geodesic flows on modular surfaces.

Contribution

It introduces a new method to compute entropy of special flows over countable Markov chains and establishes conditions for the existence of maximal entropy measures.

Findings

Derived a formula for entropy of special flows over topological Markov chains.

Provided a lower bound estimate for geodesic flow entropy on the modular surface.

Established sufficient conditions for maximal entropy measures.

Abstract

Let $Y$ be a topological Markov chain with finite leading and follower sets. Special flow over $Y$ whose height function depends on the time zero of elements of $Y$ is constructed. Then a formula for computing the entropy of this flow will be given. As an application, we give a lower estimate for the entropy of a class of geodesic flows on the modular surface. We also give sufficient conditions to guarantee the existence of a measure with maximal entropy.