Dynamics of products of nonnegative matrices

TL;DR

This paper investigates the dynamic behavior of products of nonnegative matrices, extending classical Perron-Frobenius results to complex matrix products and exploring their periodic points through advanced mathematical mappings.

Contribution

It generalizes Perron-Frobenius theory to products of multiple nonnegative matrices associated with words, including infinite sequences, and links these dynamics to exponential and logarithm maps on the positive orthant.

Findings

Extended Perron-Frobenius consequences to matrix products.

Connected periodic points of subhomogeneous maps with matrix product dynamics.

Provided a framework for analyzing infinite matrix product sequences.

Abstract

The aim of this manuscript is to understand the dynamics of products of nonnegative matrices. We extend a well known consequence of the Perron-Frobenius theorem on the periodic points of a nonnegative matrix to products of finitely many nonnegative matrices associated to a word and later to products of nonnegative matrices associated to a word, possibly of infinite length. We also make use of an appropriate definition of the exponential map and the logarithm map on the positive orthant of $\mathbb{R}^{n}$ and explore the relationship between the periodic points of certain subhomogeneous maps defined through the above functions and the periodic points of matrix products, mentioned above.