Joint spectrum, group representations, and Julia set

TL;DR

This paper explores the joint spectrum of linear operators, characterizes amenable groups via projective spectrum, and links the spectrum of certain group representations to Julia sets of rational maps, enhancing previous results.

Contribution

It provides a new characterization of amenable groups using projective spectrum and connects group representation spectra to Julia sets, improving earlier findings.

Findings

Projective spectrum characterizes amenable groups.

Spectrum of the infinite dihedral group matches Julia set of a rational map.

Enhanced understanding of self-similar group representations.

Abstract

The first half of this mostly expository note reviews some notions of joint spectrum of linear operators, and it gives a new characterization of amenable groups in terms of projective spectrum. The second half revisits an application of projective spectrum to the study of self-similar group representations made in [16]. In the case $\pi$ is the Koopman representation of the infinite dihedral group $D_\infty$ on the binary tree, it shows that the projective spectrum of $D_\infty$ coincides with the Julia set of a rational map $F_\pi: \mathbb{P}^2\to \mathbb{P}^2$ derived from the self-similarity of $\pi$. This improves the main result in [16].