Fatou-Julia dichotomy of matrix-valued polynomials

TL;DR

This paper explores the dynamics of matrix-valued polynomials in complex matrices, characterizing their Fatou and Julia sets, and constructing associated Green and Böttcher functions, extending classical complex dynamics to matrix settings.

Contribution

It provides a detailed description of Fatou and Julia sets for matrix-valued polynomials and introduces Green and Böttcher functions in this context, advancing the understanding of matrix polynomial dynamics.

Findings

Characterization of Fatou and Julia sets for matrix-valued polynomials

Construction of Green functions for matrix polynomials

Development of Böttcher-type functions for matrix polynomials

Abstract

This article gives a precise description of the Fatou sets and Julia sets of matrix-valued polynomials in $\mathcal{M}(2,\mathbb{C})$ in terms of the corresponding polynomials in $\mathbb{C}$. Further, we construct Green functions and B\"{o}ttcher-type functions for these matrix-valued polynomials.