Dynamics of Word Maps on Groups and Polynomial Maps on Algebras

TL;DR

This paper explores the dynamics of word maps on complex Lie groups and polynomial maps on associative algebras, explicitly describing their Fatou and Julia sets and analyzing their dynamical properties.

Contribution

It introduces the concepts of Fatou and Julia sets in the context of group and algebra maps and provides explicit descriptions and dynamical analysis of these sets.

Findings

Explicit descriptions of Fatou and Julia sets for the power map on GL_n(C)

No wandering Fatou components for polynomial maps on M_n(C)

Analysis of dynamics of polynomial maps induced by monic polynomials

Abstract

We introduce the notions of Fatou and Julia sets in the context of word maps on complex Lie groups and polynomial maps on finite-dimensional associative $\mathbb C$-algebras. For the group-theoretic question, we investigate the dynamics of the power map $x \mapsto x^{M}$ on the Lie group $\mathrm{GL}_n(\mathbb C)$, where $M \geq 2$ is an integer. For the algebra-related question, we study polynomial self-maps of $\mathrm{M}_n(\mathbb C)$ induced by monic polynomials in one variable. In both cases, we pin down the explicit description of the Fatou and Julia sets. We also show that there does not exist any wandering Fatou component of the pair $(p,\mathrm M_n(\mathbb C))$ where $p\in\mathbb C[z]$ is a monic polynomial of degree $\geq 2$.