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.

The spectral properties of the Frobenius-Perron operator of one-dimensional maps are studied when approaching a weakly intermittent situation. Numerical investigation of a particular family of maps shows that the spectrum becomes extremely dense and the eigenfunctions become concentrated in the vicinity of the intermittent fixed point. Analytical considerations generalize the results to a broad...

Conditions are investigated which guarantee exactness for measurable maps on measure spaces. The main application is to certain piecewise continuous maps T on [0,1] for which 7"(0) > 1. We assume [0,1] can be broken into intervals on which T is continuous and convex and at the left end of these intervals T = 0 and dt/dx > 0. Such maps have an invariant absolutely continuous density which is exact.

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...

A basic systems question concerns the concept of closure, meaning autonomomy (closed) in the sense of describing the (sub)system as fully consistent within itself. Alternatively, the system may be nonautonomous (open) meaning it receives influence from an outside coupling subsystem. Information flow, and related causation inference, are tenant on this simple concept. We take the perspective of ...

We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (In...

This paper is concerned with the data-driven optimal control of nonlinear systems. We present a convex formulation problem discounted cost function. consider problems both positive and negative discount factors. The approach relies on lifting system dynamics in space densities using linear Perron–Frobenius operator. leads to an infinite-dimensional optimization problem. approximation Koopman op...