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Symmetries of the Feinberg-Zee Random Hopping Matrix

Authors
  • Hagger, Raffael
Type
Preprint
Publication Date
Sep 03, 2015
Submission Date
Dec 05, 2014
Identifiers
DOI: 10.1142/S2010326315500161
Source
arXiv
License
Yellow
External links

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.

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