Generic properties of the lower spectral radius for some low-rank pairs of matrices

TL;DR

This paper studies the lower spectral radius of pairs of 2x2 matrices, especially when one matrix is rank one, revealing that discontinuities are prevalent and the finiteness property holds generically.

Contribution

It demonstrates that the set of discontinuities of the lower spectral radius has positive measure and analyzes the finiteness property in a simplified rank-one context.

Findings

Discontinuities form a set of positive seven-dimensional measure.

Finiteness property holds on a full measure set but not on a residual set.

Discontinuity set is large and significant in the space of matrix pairs.

Abstract

The lower spectral radius of a set of $d \times d$ matrices is defined to be the minimum possible exponential growth rate of long products of matrices drawn from that set. When considered as a function of a finite set of matrices of fixed cardinality it is known that the lower spectral radius can vary discontinuously as a function of the matrix entries. In a previous article the author and J. Bochi conjectured that when considered as a function on the set of all pairs of $2 \times 2$ real matrices, the lower spectral radius is discontinuous on a set of positive (eight-dimensional) Lebesgue measure, and related this result to an earlier conjecture of Bochi and Fayad. In this article we investigate the continuity of the lower spectral radius in a simplified context in which one of the two matrices is assumed to be of rank one. We show in particular that the set of discontinuities of the lower spectral radius on the set of pairs of $2 \times 2$ real matrices has positive seven-dimensional Lebesgue measure, and that among the pairs of matrices studied, the finiteness property for the lower spectral radius is true on a set of full Lebesgue measure but false on a residual set.