2015/10/01 by Ian D. Morris, Morris, Ian D.
Computer Science · Mathematics · #15A18 #15A60 (primary) #47D03 (secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Graph theory and applications #Matrix Theory and Algorithms #Point processes and geometric inequalities #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1510.00209
openalex publication_date 2015/10/01 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
The lower spectral radius of a set of d \× d matrices is defined to be\nthe minimum possible exponential growth rate of long products of matrices drawn\nfrom that set. When considered as a function of a finite set of matrices of\nfixed cardinality it is known that the lower spectral radius can vary\ndiscontinuously as a function of the matrix entries. In a previous article the\nauthor and J. Bochi conjectured that when considered as a function on the set\nof all pairs of 2 \× 2 real matrices, the lower spectral radius is\ndiscontinuous on a set of positive (eight-dimensional) Lebesgue measure, and\nrelated this result to an earlier conjecture of Bochi and Fayad. In this\narticle we investigate the continuity of the lower spectral radius in a\nsimplified context in which one of the two matrices is assumed to be of rank\none. We show in particular that the set of discontinuities of the lower\nspectral radius on the set of pairs of 2 \× 2 real matrices has positive\nseven-dimensional Lebesgue measure, and that among the pairs of matrices\nstudied, the finiteness property for the lower spectral radius is true on a set\nof full Lebesgue measure but false on a residual set.\n