We show how methods from nonlinear spectral theory can be used to analyse the time behaviour of dynamical discrete event systems. RÉSUMÉ. Nous montrons comment analyser le comportement temporel des systèmes à événements discrets à l’aide de résultats de théorie spectrale non-linéaire.

Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almostinvariant sets, however, often little can be said rigorously about such calculations. We attempt to...

We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint ran...

A matrix is said to have the Perron-Frobenius property if its spectral radius is an eigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such generalized nonnegative matrices are studied and certain topological aspects such as connectedness and closure are pro...

The max-plus (or tropical) algebra is obtained by replacing the addition by the maximisation (or the minimisation) and the multiplication by the addition. It arises in the dynamic programming approach to deterministic optimal control. In particular , the evolution semigroup of a first order Hamilton-Jacobi equation is linear in the max-plus sense if the Hamiltonian is convex in the adjoint vari...

Exploring long-term implications of valuation leads us to recover and use a distorted probability measure that reflects the long-term implications for risk pricing. Formally, we apply a generalized version of Perron-Frobenius theory to construct this probability measure. We discuss methods for recovering this distribution from financial market data; we apply this distribution to characterize th...

We establish a stochastic nonlinear analogue of the PerronFrobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space (Ω,F , P ) and a random mapping D(ω, ·) : R+ → R+. Under assumptions of monotonicity and homogeneity of D(ω, ·), we prove the existence of scalar and vector measurable functions α(ω) > 0 an...

In 1907, Oskar Perron showed that a positive square matrix has a unique largest positive eigenvalue with a positive eigenvector. This result was extended to irreducible nonnegative matrices by Geog Frobenius in 1912, and to irreducible nonnegative tensors and weakly irreducible nonnegative tensors recently. This result is a fundamental result in matrix theory and has found wide applications in ...