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Smoothness of Markov Partitions for Expanding Toral Endomorphisms

Chayce Hughes, Huub de Jong

Abstract

We show that an expanding toral endomorphism in dimension 2 admits a smooth (in fact linear) Markov partition if and only if some power of the corresponding integer matrix is diagonalizable with integer eigenvalues. We exhibit examples of qualitatively different smoothness behavior, and highlight the existence of a hybrid type of smoothness in dimension 2. For dimension d, we show that expanding toral endomorphisms satisfying the eigenvalue condition above admit a linear Markov partition. Finally, we provide an estimate on the Hausdorff dimension of the boundary of a Markov partition using techniques from symbolic dynamics.

Smoothness of Markov Partitions for Expanding Toral Endomorphisms

Abstract

We show that an expanding toral endomorphism in dimension 2 admits a smooth (in fact linear) Markov partition if and only if some power of the corresponding integer matrix is diagonalizable with integer eigenvalues. We exhibit examples of qualitatively different smoothness behavior, and highlight the existence of a hybrid type of smoothness in dimension 2. For dimension d, we show that expanding toral endomorphisms satisfying the eigenvalue condition above admit a linear Markov partition. Finally, we provide an estimate on the Hausdorff dimension of the boundary of a Markov partition using techniques from symbolic dynamics.

Paper Structure

This paper contains 9 sections, 15 theorems, 86 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constructing smooth Markov partitions
  4. On the hypothesis of Theorem \ref{['thm:power diag with integer eigenvals implies smooth MP']}
  5. Non-smoothness in dimension 2
  6. Real eigenvalues
  7. Complex eigenvalues
  8. Non-diagonalizable matrices
  9. Estimating Hausdorff dimension

Key Result

Theorem 1.1

A hyperbolic toral automorphism $f:\mathbb{T}^d\to\mathbb{T}^d$ induced by a matrix $A$ admits a smooth Markov partition if and only if some power of $A$ is similar over $\mathbb{Q}$ to a block diagonal matrix where each $L_i$ is a $2\times 2$ hyperbolic integer matrix with $\det L_i = \pm1$. Moreover, any smooth Markov partition for a hyperbolic toral automorphism must be linear. $\blacktriangle

Figures (4)

  • Figure 1: Iterates of $\gamma$ under a matrix $A$ with real, irrational eigenvalues of different modulus and the projection maps onto the respective eigenlines.
  • Figure 2: Iterates of $\gamma$ under a matrix $A$ with complex eigenvalues whose arguments do not lie in $\pi\mathbb{Q}$. $L=\mathbb{R}^+\vec{w}$ and $n_k$ are chosen so that $A^{n_k} \gamma^\prime(0)$ approaches the direction of $L$ counterclockwise.
  • Figure 3: Iterates of $\gamma$ under a Jordan matrix $A$. The $x$-axis is scaled down and $n_k$ are some large integers.
  • Figure 4: A Markov partition for $2102 \blacktriangleleft\blacktriangleleft$.

Theorems & Definitions (30)

  • Theorem 1.1: Cawley, 1991
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Conjecture 1.5
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 20 more