On Simultaneous Linearization

TL;DR

This paper demonstrates that under certain conditions, a linearizer at one repelling periodic point of a uniformly quasiregular map can be transformed into a linearizer at another point using translations, revealing a form of simultaneous linearization.

Contribution

It establishes conditions under which a linearizer at one periodic point can be transformed into another at a different point via translation, in the context of solutions to Schr"oder equations.

Findings

Linearizers can be related at different points via translation.

Methods involve generalized derivatives and chain rules for quasiregular maps.

Provides a framework for understanding simultaneous linearization in quasiregular dynamics.

Abstract

Given a uniformly quasiregular mapping, there is typically no reason to assume any relationship between linearizers at different repelling periodic points. However, in the current paper we prove that in the case where the uqr map arises as a solution of a Schr\"oder equation then, with some further natural assumptions, if $L$ is a linearizer at one repelling periodic point, then $L\circ T$ is a linearizer at another repelling periodic point, where $T$ is a translation. In this sense we say $L$ simultaneously linearizes $f$. In the plane, an example would be that $e^z$ simultaneously linearizes $z^2$. Our methods utilize generalized derivatives for quasiregular mappings, including a chain rule and inverse derivative formula, which may be of independent interest.