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...

I illustrate a uniied approach to the study of the decay of correlations in hyperbolic dynamical systems. The decay of correlations or, alternatively, the rate of approach of some initial distribution to an invariant one, is a widely studied problem in dynamical systems as well as in other elds. Although a full understanding is not yet available, some notable results have been obtained in the c...

In this paper, we apply the Perron-Frobenius theory for non-negative matrices to the analysis of variance asymptotics for simulations of finite state Markov chain to which importance sampling is applied. The results show that we can typically expect the variance to grow (at least) exponentially rapidly in the length of the time horizon simulated. The exponential rate constant is determined by t...

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...

In this paper, we describe a copy-and-paste method for constructing a class of infinite self-similar trees. A copy-paste tree is constructed by repeatedly attaching copies of a finite tree (called a generator) to certain designated attachment vertices. We show that each generator has an associated nonnegative matrix which can be used to determine a formula for the growth function of the copy-pa...

Precis: We discuss the behaviour of prices in an n-sector, circulating capital model with no joint production, of the type considered by Sraffa in Part I of his book. Instead of following Sraffa's approach, which uses the notion of proportions of labour to means of production in the various "layers" of means of production, we base our analysis directly on the characteristic roots and vectors of...

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...

Let x1,…,xn be points in a metric space and define the distance matrix D?Rn×n by Dij=d(xi,xj). The Perron-Frobenius Theorem implies that there is an eigenvector v?Rn with non-negative entries associated to largest eigenvalue. We prove this nearly constant sense inner product vector 1?Rn large?v,1??12??v??2??1??2 each entry satisfies vi??v??2/4n. Both inequalities are sharp.