Spectral theory for transfer operators on compact quotients of Euclidean buildings

TL;DR

This paper extends geodesic flow concepts to Euclidean building quotients, analyzing transfer operators' spectra to understand their joint spectral properties in a new geometric setting.

Contribution

It introduces a multiparameter flow on Euclidean building quotients and studies the spectral behavior of associated transfer operators, revealing their joint spectrum characteristics.

Findings

Outside small neighborhoods of zero, the Taylor spectrum is contained in the joint point spectrum.

The work generalizes geodesic flow to a broader class of geometric structures.

Spectral analysis is performed on Lipschitz spaces for transfer operators.

Abstract

In this paper we generalize the geodesic flow on (finite) homogeneous graphs to a multiparameter flow on compact quotients of Euclidean buildings. Then we study the joint spectra of the associated transfer operators acting on suitable Lipschitz spaces. The main result says that outside an arbitrarily small neighborhood of zero in the set of spectral parameters the Taylor spectrum of the commuting family of transfer operators is contained in the joint point spectrum.