Bowen Measure From Heteroclinic Points

TL;DR

This paper introduces a novel method for constructing the Bowen measure on Smale spaces using heteroclinic points, extending Bowen's periodic point approach through direct computation and factor maps.

Contribution

It provides a new heteroclinic point-based construction of the Bowen measure and analyzes its growth and component structure, broadening understanding of invariant measures on Smale spaces.

Findings

Constructed Bowen measure using heteroclinic points.

Proved growth rate results for heteroclinic sets.

Extended results from shifts of finite type to general Smale spaces.

Abstract

We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another: heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.