Removing quasiconformal orbits

TL;DR

This paper proves that generic continuous linear cocycles over zero-dimensional systems lack quasiconformal orbits, using a novel tower construction for zero-dimensional homeomorphisms, partially answering a question in the field.

Contribution

It introduces a new tower construction for zero-dimensional homeomorphisms and demonstrates the absence of quasiconformal orbits in generic cocycles, addressing an open question.

Findings

Generic continuous linear cocycles over zero-dimensional systems have no quasiconformal orbits.

A new tower construction for zero-dimensional homeomorphisms is developed.

The results partially answer a question posed by Nassiri, Rajabzadeh, and Reshadat.

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.