Smoothness of Markov Partitions for Expanding Toral Endomorphisms

TL;DR

This paper investigates conditions under which expanding toral endomorphisms admit smooth or linear Markov partitions, providing criteria, examples, and Hausdorff dimension estimates for their boundaries.

Contribution

It establishes a precise eigenvalue condition for smooth Markov partitions in dimension 2 and extends results to higher dimensions, also analyzing boundary complexity.

Findings

A smooth (linear) Markov partition exists iff some power of the matrix is diagonalizable with integer eigenvalues.

Examples demonstrate different smoothness behaviors and a hybrid smoothness type in dimension 2.

An estimate on the Hausdorff dimension of the boundary of Markov partitions is provided.

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.