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

Gregory Hemenway

TL;DR

The paper addresses turning equilibrium states for the model expanding map $E_d$ on $\mathbb{T}^n$ into Lebesgue measure via a conjugacy to a Lebesgue-invariant expanding skew product. It develops a nonstationary, fiberwise transfer-operator framework and leverages Markov partitions to count conjugacies, thereby generalizing McMullen’s circle results to higher dimensions without Teichmüller theory. The main achievement is proving the existence of a $C^{1+\alpha}$ expanding skew product $F$ conjugate to $E_d$ such that $\mu$ is pushed forward to Lebesgue measure, with exactly $d$ orientation-preserving conjugacies, extended to all $n\ge 2$ by induction. This advances the understanding of how equilibrium states relate to Lebesgue measure in high-dimensional expanding dynamics and provides a Teichmüller-free combinatorial count of conjugacies via Markov partitions.

Abstract

We show that any equilibrium state for a Hölder 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 $\mathbb{T}^n$.

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

TL;DR

The paper addresses turning equilibrium states for the model expanding map on into Lebesgue measure via a conjugacy to a Lebesgue-invariant expanding skew product. It develops a nonstationary, fiberwise transfer-operator framework and leverages Markov partitions to count conjugacies, thereby generalizing McMullen’s circle results to higher dimensions without Teichmüller theory. The main achievement is proving the existence of a expanding skew product conjugate to such that is pushed forward to Lebesgue measure, with exactly orientation-preserving conjugacies, extended to all by induction. This advances the understanding of how equilibrium states relate to Lebesgue measure in high-dimensional expanding dynamics and provides a Teichmüller-free combinatorial count of conjugacies via Markov partitions.

Abstract

We show that any equilibrium state for a Hölder potential on the model map on is conjugate to Lebesgue measure for an invariant expanding skew product of degree . 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 .
Paper Structure (10 sections, 13 theorems, 35 equations, 4 figures)

This paper contains 10 sections, 13 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Existence and Uniqueness of Equilibrium States
  4. Semiconjugacies of Expanding Maps on $\mathbbm{S}^1$
  5. Statements of Main Results
  6. Skew Products
  7. Prior Results
  8. Conjugacies on $\mathbbm{T}^2$
  9. Construction of conjugacy on $\mathbbm{T}^2$
  10. Extension to conjugacies on $\mathbbm{T}^n$

Key Result

Theorem A

Let $\mu_{\varphi}$ be an equilibrium state for $(\mathbbm{T}^2,E_d)$ for a Hölder potential $\varphi$ of exponent $\alpha$. There exists a conjugacy $H\colon \mathbbm{T}^2\to\mathbbm{T}^2$ such that Lebesgue is preserved by a $C^{1+\alpha}$ expanding $F=H\circ E_d\circ H^{-1}$.

Figures (4)

  • Figure 1: Commutative diagram between $(\mathbbm{S}^1,E_d)$ and $(\mathbbm{S}^1,f)$.
  • Figure 2: Commutative diagram between $(Y_x,F_x)$ and $(Y_x,g_x)$.
  • Figure 3: The homeomorphisms $H_X$ and $H_Y$ reorganizing fibers according to the dynamics of $F$.
  • Figure 4: Commutative diagram between $(\mathbbm{T}_x,G_x)$ and $(\mathbbm{T}^{n-1},E_d)$.

Theorems & Definitions (20)

  • Theorem A
  • Theorem 2.1: Variational Principle
  • Theorem 2.2: RPF Theorem (See B08 & W78)
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • Theorem 3.1: Hemenway H23
  • Lemma 4.1
  • proof
  • ...and 10 more