The problem of phase space transport, which is of interest from both the theoretical and practical point of view, has been investigated extensively using geometric and probabilistic methods. Two important tools to study this problem that have emerged in recent years are finite-time Lyapunov exponents (FTLE) and the Perron–Frobenius operator. The FTLE measures the averaged local stretching aroun...

In this contribution we propose a technique to analyse arbitrary invariant subsets of chaotic dynamical systems. For this purpose we introduce the constrained Frobenius-Perron operator. We demonstrate the use of this operator by determining the geometrical multifractal spectrum of invariant chaotic subsets of one-dimensional maps which are either coexisting side by side indepen­ dently or are e...

We develop a resummed high-temperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the Perron-Frobenius operator of weakly coupled map lattices.

We discuss spectral representations of the Perron-Frobenius operator, U, associated with a highly chaotic map. The continuous spectrum of U does not contain (except coincidentally) information about physically accessible quantities such as decay rates of correlation functions. We show constructively that decay rates can be incorporated into a generalized spectral decomposition of U if its domai...

We develop a resummed high-temperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the Perron-Frobenius operator of weakly coupled map lattices.

We give a new proof of a result due to Ruelle about the existence and simplicity of a unique maximal eigenvalue for a Ruelle^Perron^Frobenius operator acting on some Ho« lder continuous function space. Mathematics Subject Classi¢cations (1991): Primary 58F23, Secondary 30C62. Key words: locally expanding, mixing, Ruelle^Perron^Frobenius operator, maximal eigenvalue. 1. Introduction Ruelle's The...

To study the convergence to equilibrium in random maps, we develop the spectral theory of the corresponding transfer (Perron— Frobenius) operators acting in a certain Banach space of generalized functions (distributions). The random maps under study in a sense fill the gap between expanding and hyperbolic systems, since among their (deterministic) components there are both expanding and contrac...

This note presents a new proof of basic Perron{Frobenius theory of irreducible nonnegative matrices. ABSTRACT This note presents a new proof of basic Perron{Frobenius theory of irreducible nonnegative matrices.