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Spectral theory for transfer operators on compact quotients of Euclidean buildings

Joachim Hilgert, Daniel Kahl, Tobias Weich

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.

Spectral theory for transfer operators on compact quotients of Euclidean buildings

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.

Paper Structure

This paper contains 26 sections, 63 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General Setting
  3. Simplicial Complexes
  4. Euclidean and Spherical Coxeter Complexes
  5. Coxeter Systems
  6. Root Systems
  7. Coroots and Coweights
  8. Euclidean and Spherical Buildings
  9. Buildings
  10. Morphisms and their Properties
  11. Dynamics on the Space of Sectors
  12. Sectors in Local Buildings
  13. Ultrametric on Sectors
  14. Shift Operators
  15. Transfer Operator
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a compact local building. Then, for any $0 < \vartheta < 1$, we have where we define

Figures (3)

  • Figure 1: Non-backtracking path
  • Figure :
  • Figure :

Theorems & Definitions (205)

  • Theorem 1.1
  • Definition 2.1: Simplicial structure
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: Simplicial Substructure
  • Definition 2.5: Morphisms of Simplicial Structures
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8: Labeling for Simplicial Structures
  • Definition 2.9: Labeled Simplicial Substructures
  • ...and 195 more