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

The lower spectral radius of a set of d d matrices is de ned to be the minimum possible exponential growth rate of long products of matrices drawn from that set. When considered as a function of a nite set of matrices of xed 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 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 simpli ed 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 2 real matrices has positive seven-dimensional Lebesgue measure, and that among the pairs of matrices studied, the niteness property for the lower spectral radius is true on a set of full Lebesgue measure but false on a residual set