Symmetries of the Feinberg-Zee Random Hopping Matrix

TL;DR

This paper investigates the symmetries of the spectrum of the Feinberg-Zee Random Hopping Matrix, revealing an infinite sequence of symmetries and Julia sets that contribute to understanding its fractal-like boundary behavior.

Contribution

It identifies an infinite sequence of spectral symmetries and demonstrates the presence of Julia sets within the spectrum, expanding understanding of its structure.

Findings

Spectrum invariant under square roots

Unit disk contained in the spectrum

Presence of Julia sets within the spectrum

Abstract

We study the symmetries of the spectrum of the Feinberg-Zee Random Hopping Matrix. Chandler-Wilde and Davies proved that the spectrum of the Feinberg-Zee Random Hopping Matrix is invariant under taking square roots, which implied that the unit disk is contained in the spectrum (a result already obtained slightly earlier by Chandler-Wilde, Chonchaiya and Lindner). In a similar approach we show that there is an infinite sequence of symmetries at least in the periodic part of the spectrum (which is conjectured to be dense). Using these symmetries, we can exploit a considerably larger part of the spectrum than the unit disk. As a further consequence we find an infinite sequence of Julia sets contained in the spectrum. These facts may serve as a part of an explanation of the seemingly fractal-like behaviour of the boundary.