On the Spectrum of Hecke Type Operators related to some Fractal Groups

TL;DR

This paper presents the first example of a connected 4-regular graph with a Cantor set spectrum for its Laplace operator, linking fractal groups, Julia sets, and Hecke operators through spectral analysis.

Contribution

It introduces a novel example of a graph with a Cantor spectrum, connecting fractal groups, Julia sets, and Hecke operators, and develops a finite approximation method for spectral computation.

Findings

Spectrum of the graph's Laplace operator is a Cantor set.

Spectra are related to Julia sets of quadratic maps.

Self-similarity is reflected in operator approximations.

Abstract

We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common ``finite approximation'' method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also ``substitutional graphs''. We also formulate our results in terms of Hecke type operators related to some irreducible quasi-regular representations of fractal groups and in terms of the Markovian operator associated to noncommutative dynamical systems via which these fractal groups were originally defined. In the computations we performed, the self-similarity of the groups is reflected in the self-similarity of some operators; they are approximated by finite counterparts whose spectrum is computed by an ad hoc factorization process.