A class of Latt\`es maps with cellular structures

TL;DR

This paper introduces orthotopic Lattès maps, a class of quasiregular maps with cellular structures, expanding Thurston-type dynamics, and visual metrics equivalent to Riemannian distances, enriching the understanding of quasiregular dynamics.

Contribution

It demonstrates that orthotopic Lattès maps are cellular Markov maps, providing new examples of expanding Thurston-type maps with specific geometric and dynamical properties.

Findings

Orthotopic Lattès maps are cellular Markov maps.

These maps are expanding Thurston-type maps.

Visual metrics are quasisymmetrically equivalent to Riemannian distances.

Abstract

We show that a class of quasiregular Latt\`es maps, called orthotopic Latt\`es maps, are cellular Markov maps. This provides examples of expanding Thurston-type maps that are also uniformly quasiregular, and whose visual metrics are quasisymmetrically equivalent to the Riemannian distance.