Work

The joint spectrum

Year: 2020

Type: article

Abstract: We introduce the notion of joint spectrum of a compact set of matrices S⊂GLd⁡(C), which is a multi-dimensional generalization of the joint spectral radius. We begin with a thorough study of its properties (under various assumptions: irreducibility, Zariski-density, and domination). Several classical properties of the joint spectral radius are shown to hold in this generalized setting and an analogue of the Lagarias–Wang finiteness conjecture is discussed. Then we relate the joint spectrum to matrix valued random processes and study what points of it can be realized as Lyapunov vectors. We also show how the joint spectrum encodes all word metrics on reductive groups. Several examples are worked out in detail. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.

Source: Journal of the London Mathematical Society

DOI: 10.1112/jlms.12397

Authors: Emmanuel Breuillard, Cagri Sert

Institutions: University of Cambridge, ETH Zurich

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung, European Research Council

Awards: Equidistribution and dynamics on homogeneous spaces, G4171367792