ИСТИНА

The Perron-Frobenius theory relates the combinatorial, spectral, and dynamical properties of a nonnegative matrix. In particular, it ensures the convergence of a Markov chain in terms of the graph of its transfer matrix. We consider possible generalizations of this theory to an arbitrary multiplicative semigroup of nonnegative matrices. We show that it is possible to characterize primitive and scrambling (in various senses) semigroups in terms of their spectra and of their multigraphs. This, in particular, leads to direct extensions of some well-known facts and notions of the Perron-Frobenius theory (imprimitivity index, Romanovsky's theorem, etc.) to matrix semigroups. Several applications to non-homogeneous and multivariate Markov chains, to dynamical systems, synchronizing automata, etc. will be discussed.