The Spectrum of the Quaquaversal Operator is Real

TL;DR

This paper proves that the spectrum of the Hecke operator related to the Quaquaversal tiling is real and provides explicit calculations for most of its eigenvalues, addressing a previously open question.

Contribution

It establishes the reality of the spectrum for the Quaquaversal Hecke operator and explicitly computes a significant portion of its eigenvalues, advancing understanding of its spectral properties.

Findings

The spectrum of the Hecke operator is real.

Approximately 75% of the eigenvalues are explicitly computed.

Addresses an open question by Draco, Sadun, and Van Wieren.

Abstract

The Hecke operator associated with the Quaquaversal tiling, a highly anisotropic tiling introduced by Conway and Radin, is shown to have a real spectrum. Answering a question of Draco, Sadun, and Van Wieren. We also explicitly compute about three quarters of this operator's eigenvalues.