Continuity properties of the lower spectral radius

Abstract

The lower spectral radius, or joint spectral subradius, of a set of real d × 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 × d matrices. As an application we exhibit open sets of pairs of 2 × 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.

On Falconer's formula for the generalised Rényi dimension of a self-affine measure

“…Indeed, the examples of discontinuity of r q in Theorem 1.3(i) and (ii) correspond directly with known examples of the discontinuity of the lower spectral radius, specifically Example 1.1 and Proposition 7.6 in [7]. In the context of Theorem 1.3(i) we can obtain sufficient control on the size of the discontinuity in ̺ without any assumptions on p and q.…”

“…The lower spectral radius is known to depend discontinuously on the matrix entries in general [20, p. 20], and this phenomenon was investigated in depth by the author and Bochi in [7]. This relates to R q (A 1 , A 2 , p, s) as follows: if 0 < s ≤ 1 then we may estimate…”

“…In the context of Theorem 1.3(ii) our much weaker control on the discontinuities of ̺ means that the above estimate is only useful if q is large and p is close to 1 2 , which has the effect of bringing the quantities log(p q + (1 − p) q ) and q log(min{p, (1 − p)}) closer together. In order to deal with more general p the proof of Theorem 1.3(ii) in fact applies a slightly finer estimate than that indicated above: for this we require a slightly strengthened statement of [7,Proposition 7.6], which shows not only that the lower spectral radius is discontinuous in certain places but also specifies how it is discontinuous. We nonetheless emphasise that the conceptual origin of the discontinuity of r q in this paper is that it is a consequence of the discontinuity of the lower spectral radius.…”

“…Bonatti has constructed explicit examples of resistant pairs in which A 2 is a rational rotation, and these examples are described in [7].…”

Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

“…The first explicit counterexample to the finiteness conjecture was built in [16], while the general methods of constructing of such a type of counterexamples were presented recently in [17,18]. The lower radius in a sense is more complex object for analysis than the generalized spectral radius because it generally depends on A not continuously [19,20]. However, for the lower spectral radius, disproof of the finiteness conjecture was found to be even easier [12,21] than for the generalized spectral radius.…”