We investigate dynamical systems with stochastic perturbation and study to what extend analytical properties of the noise present influence the spectrum of the associated Frobenius–Perron operator. We suggest to distinguish a “physical” part of the spectrum of the deterministic system, as this robust with respect to the perturbation. For exemplary system studied such eigenvalues of the FP-opera...

In this note we give a computable criterion for a piecewise expanding interval map T to be mixing, which at the same time not only establishes explicit bounds on the spectral gap of the associated Perron Frobenius operator acting on the space of functions of bounded variation, but establishes strict contraction rates for this operator. Of course such a result cannot be completely general, but o...

We describe a fast and accurate method to compute the pressure and equilibrium states for maps of the interval T:[0,1]-->[0,1] with respect to potentials phi:[0,1]-->R. An approximate Ruelle-Perron-Frobenius operator is constructed and the pressure read off as the logarithm of the leading eigenvalue of this operator. By setting phi identical with 0, we recover the topological entropy. The confo...

We discuss the existence of large isolated (non-unit) eigenvalues of the Perron– Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or ‘resonances’) are distributions which approach the invariant density (or equilibrium distribution) at a rate slower than that prescribed by the minimal expansion rate. We consider the transitional behaviour of the eigenfunctions ...

We study the Perron-Frobenius operator P of closed dynamical systems and certain open dynamical systems. We prove that the presence of a large positive eigenvalue ρ of P guarantees the existence of a 2-partition of the phase space for which the escape rates of the open systems defined on the two partition sets are both slower than − log ρ. The open systems with slow escape rates are easily iden...

Let V be a vector space over a fully ordered field F. In Sec. 2 we characterize cones K with ascending chain condition (ACC) on faces of Ii. In Sec. 3 we show that if K has ACC on faces, then an operator A is strongly irreducible if and only if A is irreducible. In Sec. 4 we prove theorems of Perron-Frobenius type for a strongly irreducible operator A in the case that FR, the real field, and K ...

In this paper, we study random Blaschke products, acting on the unit circle, and consider cocycle of Perron–Frobenius operators Banach spaces analytic functions an annulus. We completely describe Lyapunov spectrum these cocycles. As a corollary, obtain simple product system where has infinitely many distinct exponents, but arbitrarily small natural perturbations cause complete collapse spectrum...

Recently, Storm [10] defined the Ihara-Selberg zeta function of a hypergraph, and gave two determinant expressions of it by the Perron-Frobenius operator of a digraph and a deformation of the usual Laplacian of a graph. We present a new determinant expression for the Ihara-Selberg zeta function of a hypergraph, and give a linear algebraic proof of Storm’s Theorem. Furthermore, we generalize the...