具有常谱半径的矩阵半群
Matrix semigroups with constant spectral radius
摘要：
具有常谱半径（c.s.r.）的乘法矩阵半群研究并应用于代数、组合数学、函数方程和动力系统的若干问题。我们证明了所有这类半群都是用不可约半群刻画的。每个不可约的c.s.r.半群定义了欧氏球面上的游动，它的所有非奇异元素都类似于（在同一基上）正交。我们把所有非阴性的c.s.r.半群和任意低维半群。对于更高的维度，我们描述了五个类，并留下一个开放的问题，该列表的完整性。c.s.r.的算法识别问题。性质对于不可约半群是多项式可解的，而对于可约半群是不可判定的。
英文摘要：
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.
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