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

Abstract

We show that any equilibrium state for a Hölder potential on the model map $\vec{x}\mapsto d\cdot \vec{x}\mathrm{mod}{\mathbb{Z}}^n$ 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 ${\mathbb{T}}^n$.

14 pages, 4 figures