Perron-Frobenius theorem and invariant sets in linear systems dynamics

The paper explores the connections between the Perron-Frobenius (PF) theory and the flow-invariant sets with respect to the dynamics of linear systems. Our analysis includes both discrete-and continuous-time systems, and the results are separately formulated for linear dynamics generated by the following types of matrices: (i) (essentially) nonnegative and irreducible or (essentially) positive, (ii) (essentially) nonnegative and reducible. For both cases we show how the PF eigenvalue and right and left PF eigenvectors are related to invariant sets defined for any Holder p-norm (1 les p les infin ).