Conjugacies of Expanding Skew Products on $\mathbb{T}^n$

TL;DR

This paper generalizes McMullen's result to higher dimensions, showing that equilibrium states for certain expanding skew products on tori are conjugate to Lebesgue measure, using transfer operators and Markov partitions.

Contribution

It extends the classification of equilibrium states to higher-dimensional tori for expanding skew products, employing a novel approach with nonstationary transfer operators.

Findings

Equilibrium states are conjugate to Lebesgue measure.

Classification of invariant measures for expanding maps on $\

Abstract

We show that any equilibrium state for a H\"older potential on the model map $\vec{x} \mapsto d \cdot \vec{x} \mod \mathbb{Z}^n$ on $\mathbb{T}^n$ is conjugate to Lebesgue measure for an invariant expanding skew product of degree $d$. This is a generalization of a result of McMullen to higher dimensions for equilibrium states. We use an approach developed by the author using a family of nonstationary transfer operators for an expanding skew product. We also apply a Markov partition argument to classify invariant probability measures for expanding maps on $\mathbb{T}^n$.