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Removing quasiconformal orbits

Jairo Bochi

TL;DR

This paper shows that for zero-dimensional compact spaces X with a homeomorphism T possessing finitely many periodic points per period, generic continuous linear cocycles over T do not admit quasiconformal orbits. The proof builds a novel tower-based framework in topological dynamics, introducing an $N$-capturing tower lemma and assembling castles that control periodic and nonperiodic orbits simultaneously. Using these structures, the author constructs precise perturbations of cocycles on tower pieces to force unbounded distortion along some iterate, proving that the set of cocycles with quasiconformal orbits is meager in $C^0(X,GL(d,\mathbb{F}))$ (with $d\ge2$, and $d\ge3$ when $\mathbb{F}=\mathbb{R}$ and periodic points exist). The approach yields a robust, purely topological mechanism (towers and castles) to eliminate bounded distortion and provides a potentially independent interest result on towers in zero-dimensional dynamics, applicable to subshifts on finite alphabets.

Abstract

We show that generic continuous linear cocycles over shifts and other zero-dimensional systems admit no quasiconformal orbits, thus providing a partial answer to a question of Nassiri, Rajabzadeh, and Reshadat. The proof relies on a new result about towers for homeomorphisms of zero-dimensional spaces, which may be of independent interest.

Removing quasiconformal orbits

TL;DR

This paper shows that for zero-dimensional compact spaces X with a homeomorphism T possessing finitely many periodic points per period, generic continuous linear cocycles over T do not admit quasiconformal orbits. The proof builds a novel tower-based framework in topological dynamics, introducing an -capturing tower lemma and assembling castles that control periodic and nonperiodic orbits simultaneously. Using these structures, the author constructs precise perturbations of cocycles on tower pieces to force unbounded distortion along some iterate, proving that the set of cocycles with quasiconformal orbits is meager in (with , and when and periodic points exist). The approach yields a robust, purely topological mechanism (towers and castles) to eliminate bounded distortion and provides a potentially independent interest result on towers in zero-dimensional dynamics, applicable to subshifts on finite alphabets.

Abstract

We show that generic continuous linear cocycles over shifts and other zero-dimensional systems admit no quasiconformal orbits, thus providing a partial answer to a question of Nassiri, Rajabzadeh, and Reshadat. The proof relies on a new result about towers for homeomorphisms of zero-dimensional spaces, which may be of independent interest.

Paper Structure

This paper contains 6 sections, 8 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Towers and castles
  3. Capturing property and periodic orbits
  4. Castles with capturing towers
  5. Eradicating quasiconformal orbits
  6. Further comments

Key Result

theorem 1.1

Let $X$ be a compact metric space of zero topological dimension. Let $T \colon X \to X$ be a homeomorphism with finitely many periodic points of any given period. Let $\mathbb{F}$ be either $\mathbb{C}$ or $\mathbb{R}$, and let $d \ge 2$. Additionally, assume that $d \ge 3$ if $\mathbb{F} = \mathbb{

Figures (2)

  • Figure 1: An $N$-capturing neighborhood $B$ of a hyperbolic fixed point. If $B$ is taken small enough, then the complement $B^\mathsf{c}$ is also $N$-capturing.
  • Figure 2: The two castles $K$ and $L$. The arrows denote possible excursions outside of $K\sqcup L$. Since the set $K^\mathsf{c}$ is $N$-capturing, excursions following the leftmost arrow take time at least $N$.

Theorems & Definitions (23)

  • theorem 1.1
  • remark 1.2
  • remark 1.3
  • definition 2.1
  • definition 2.2
  • example 2.3
  • lemma 2.4
  • proof
  • lemma 2.5
  • proof
  • ...and 13 more