Matrix semigroups with constant spectral radius

TL;DR

This paper characterizes matrix semigroups with constant spectral radius, classifies nonnegative and low-dimensional cases, and explores algorithmic recognition, revealing polynomial solvability for irreducible cases and undecidability for reducible ones.

Contribution

It provides a comprehensive classification of c.s.r. semigroups, characterizes their structure via irreducibility, and analyzes the computational complexity of recognizing such semigroups.

Findings

All irreducible c.s.r. semigroups are similar to orthogonal matrices.

Classification of nonnegative and low-dimensional c.s.r. semigroups is achieved.

Recognition of c.s.r. property is polynomial for irreducible and undecidable for reducible semigroups.

Abstract

Multiplicative matrix semigroups with constant spectral radius (c.s.r.) are studied and applied to several problems of algebra, combinatorics, functional equations, and dynamical systems. We show that all such semigroups are characterized by means of irreducible ones. Each irreducible c.s.r. semigroup defines walks on Euclidean sphere, all its nonsingular elements are similar (in the same basis) to orthogonal. We classify all nonnegative c.s.r. semigroups and arbitrary low-dimensional semigroups. For higher dimensions, we describe five classes and leave an open problem on completeness of that list. The problem of algorithmic recognition of c.s.r. property is proved to be polynomially solvable for irreducible semigroups and undecidable for reducible ones.