Conjugacies of Expanding Skew Products on \mathbbTⁿ

2025/05/14 by Gregory Hemenway, Hemenway, Gregory
Mathematics · #37D35 (Primary) 37C30 (Secondary) #Algebraic and Geometric Analysis #Dynamical Systems (math.DS) #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2505.09575

openalex publication_date 2025/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that any equilibrium state for a Hölder potential on the model map x ↦ d ⋅ x \mod ℤⁿ on \mathbbTⁿ 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 \mathbbTⁿ.

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