General harmonic measures for distance-expanding dynamical systems

TL;DR

This paper develops a framework for harmonic measures in distance-expanding dynamical systems, connecting random walks on associated hyperbolic graphs with fractal dimensions of measures on the system.

Contribution

It introduces a novel approach to harmonic measures for distance-expanding systems using Gromov hyperbolic tile graphs and explores their boundary behavior and dimensional properties.

Findings

Martin boundary admits a surjection to the space X

Harmonic measure's fractal dimension equals asymptotic random walk quantities

Surjection from Martin boundary to X may not be a homeomorphism

Abstract

Partially motivated by the study of I. Binder, N. Makarov, and S. Smirnov [BMS03] on dimension spectra of polynomial Cantor sets, we initiate the investigation on some general harmonic measures, inspired by Sullivan's dictionary, for distance-expanding dynamical systems. Let $f\colon X\to X$ be an open distance-expanding map on a compact metric space $(X,\rho)$. A Gromov hyperbolic tile graph $\Gamma$ associated to the dynamical system $(X,f)$ is constructed following the ideas from M. Bonk, D. Meyer [BM17] and P. Ha\"issinsky, K. M. Pilgrim [HP09]. We consider a class of one-sided random walks associated with $(X,f)$ on $\Gamma$. They induce a Martin boundary of the tile graph, which may be different from the hyperbolic boundary. We show that the Martin boundary of such a random walk admits a surjection to $X$. We provide a class of examples to show that the surjection may not be a homeomorphism. Such random walks also induce measures on $X$ called harmonic measures. When $\rho$ is a visual metric, we establish an equality between the fractal dimension of the harmonic measure and the asymptotic quantities of the random walk.