Continuity properties of the lower spectral radius

Abstract

The lower spectral radius, or joint spectral subradius, of a set of real $d\times d$ matrices is defined to be the smallest possible exponential growth rate of long products of matrices drawn from that set. The lower spectral radius arises naturally in connection with a number of topics including combinatorics on words, the stability of linear inclusions in control theory, and the study of random Cantor sets. In this article we apply some ideas originating in the study of dominated splittings of linear cocycles over a dynamical system to characterise the points of continuity of the lower spectral radius on the set of all compact sets of invertible $d\times d$ matrices. As an application we exhibit open sets of pairs of $2\times 2$ matrices within which the analogue of the Lagarias-Wang finiteness property for the lower spectral radius fails on a residual set, and discuss some implications of this result for the computation of the lower spectral radius.

Final version, revised according to the referee reports

References in corpus (2)

• Mather sets for sequences of matrices and applications to the study of joint spectral radii
• Undecidability in binary tag systems and the Post correspondence problem for four pairs of words

Cited by in corpus (14)

• Ergodic optimization in dynamical systems
• The Berger-Wang formula for the Markovian joint spectral radius
• The joint spectrum
• Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices
• Finer geometry of planar self-affine sets
• Parameter spaces of locally constant cocycles
• Slices of the Takagi function
• Constructive stability and stabilizability of positive linear discrete-time switching systems
• Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems
• Non-invertible planar self-affine sets
• Multicones, duality and invariance for families of matrices
• Fast approximation of the $p$-radius, matrix pressure or generalised Lyapunov exponent for positive and dominated matrices
• Large deviation principles for non-elementary random walks on hyperbolic spaces
• Robust transitivity and domination for endomorphisms displaying critical points