Dynamics of Word Maps on Groups and Polynomial Maps on Algebras

TL;DR

This work studies the dynamics of word maps on complex Lie groups and polynomial maps on matrix algebras through Fatou and Julia sets. It reduces the matrix dynamics to spectral data via Jordan form, establishing that a matrix lies in the Fatou set of a polynomial map precisely when its spectrum lies in the scalar Fatou set, and proving that $\mathscr{F}_{x^{M}}(GL_n(\mathbb C)) = \mathscr{F}_{x^{M}}(\mathrm{M}_n(\mathbb C)) \cap GL_n(\mathbb C)$. A corollary shows there are no wandering Fatou components for monic polynomials of degree $\ge 2$ on $\mathrm{M}_n(\mathbb C)$. By linking several complex variables dynamics to matrix dynamics, the results yield explicit descriptions of Fatou/Julia sets in these noncommutative settings and provide a spectral criterion for stability.

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 $\geq 2$.

Theorems & Definitions (20)

• theorem 1
• Definition 2.1
• Definition 2.2
• Example 2.3
• Example 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1
• proof
• Remark 3.2