The joint spectrum

Abstract

We introduce the notion of \emph{joint spectrum} of a compact set of matrices $S\subset G{L}_d(\mathbb{C})$, 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, 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.

52 pages, 6 figures. v3: minor changes

References in corpus (9)

• Weakly commensurable arithmetic groups, lengths of closed geodesics and isospectral locally symmetric spaces
• Anosov representations and dominated splittings
• Mather sets for sequences of matrices and applications to the study of joint spectral radii
• Continuity properties of the lower spectral radius
• Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles
• Inequalities for numerical invariants of sets of matrices
• An explicit Lipschitz constant for the joint spectral radius
• On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
• Quasi-multiplicativity of typical cocycles

Cited by in corpus (1)

• The finiteness conjecture holds in SL(2,Z>=0)^2