Poincare linearizers in higher dimensions

TL;DR

This paper extends the concept of Poincaré linearizers to higher dimensions, analyzing their dynamics and showing that their fast escaping sets form a spider's web structure.

Contribution

It introduces a higher-dimensional generalization of Poincaré linearizers and studies their dynamical properties, including the structure of their escaping sets.

Findings

Fast escaping set has a spider's web structure

Higher-dimensional linearizers conjugate quasiregular mappings to linear maps

Extension of classical complex dynamics to higher dimensions

Abstract

It is well-known that a holomorphic function near a repelling fixed point may be conjugated to a linear function. The function which conjugates is called a Poincar\'e linearizer and may be extended to a transcendental entire function in the plane. In this paper, we study the dynamics of a higher dimensional generalization of Poincar\'e linearizers. These arise by conjugating a uniformly quasiregular mapping in $\R^m$ near a repelling fixed point to the mapping $x\mapsto 2x$. In particular, we show that the fast escaping set of such a linearizer has a spider's web structure.