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A class of Lattès maps with cellular structures

Zhiqiang Li, Hanyun Zheng

TL;DR

The paper identifies a class of higher-dimensional Lattès maps, called orthotopic Lattès maps, and proves they are cellular Markov maps. By constructing symmetric cube decompositions tied to orthotopic crystallographic groups, it builds cellular Markov partitions that render chaotic cases expanding Thurston-type maps with visual metrics quasisymmetric to the ambient Riemannian distance. This solidifies a Thurston-type framework for a broad family of uniformly quasiregular Lattès maps in dimensions n≥3. The approach clarifies when Lattès dynamics exhibit cellular, Markov, and quasisymmetric geometric features in a higher-dimensional setting.

Abstract

We show that a class of quasiregular Lattès maps, called orthotopic Lattès 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.

A class of Lattès maps with cellular structures

TL;DR

The paper identifies a class of higher-dimensional Lattès maps, called orthotopic Lattès maps, and proves they are cellular Markov maps. By constructing symmetric cube decompositions tied to orthotopic crystallographic groups, it builds cellular Markov partitions that render chaotic cases expanding Thurston-type maps with visual metrics quasisymmetric to the ambient Riemannian distance. This solidifies a Thurston-type framework for a broad family of uniformly quasiregular Lattès maps in dimensions n≥3. The approach clarifies when Lattès dynamics exhibit cellular, Markov, and quasisymmetric geometric features in a higher-dimensional setting.

Abstract

We show that a class of quasiregular Lattès maps, called orthotopic Lattès 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.

Paper Structure

This paper contains 8 sections, 21 theorems, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasiregular maps and Lattès maps
  4. Cellular Markov maps
  5. Orthotopic Lattès maps
  6. Orthotopic crystallographic groups and Lattès triples
  7. Symmetry group and symmetric decomposition of cube
  8. Cellular Markov partitions of orthotopic Lattès maps

Key Result

Theorem A

An orthotopic Lattès map $\mathcal{M}^n \to \mathcal{M}^n$, $n\geqslant3$, on a connected closed Riemannian $n$-manifold $\mathcal{M}^n$ is a cellular Markov map. Moreover, if $f$ has topological degree at least two, then $f$ is a Thurston-type map.

Figures (4)

  • Figure 3.1: $H^{2}_{22}(1,-1)$, $H^2_{12}(1,1)$, and $\mathcal{K}_{2}$.
  • Figure 3.2: Illustration of Lemmas \ref{['l: orthotopic Lattes: property of K^o_n']} (left) and \ref{['l: orthotopic Lattes: K_n-1 in dK_n']} (right).
  • Figure 3.3: $\mathcal{K}_2$, $\mathcal{K}_{2,2}$, and $\mathcal{K}_{2,3}$.
  • Figure 3.4:

Theorems & Definitions (47)

  • Theorem A
  • Corollary
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3: Cell decompositions
  • Lemma 2.4: BM17
  • Lemma 2.5: LPZ25
  • Definition 2.6: Refinements
  • Lemma 2.7: BM17
  • Lemma 2.8: LPZ25
  • ...and 37 more