  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.
