We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the Perron-Frobenius operator of an expanding map T of [0, 1] with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of f ◦ T , when f belongs to a large class of unbounded functions from [0, 1] to R. We also prove other limit theorems and momen...

This manuscript introduces a regression-type formulation for approximating the Perron-Frobenius Operator by relying on distributional snapshots of data. These may represent densities particles. The Wasserstein metric is leveraged to define suitable functional optimization in space distributions. allows seeking dynamics so as interpolate flow function space. A first-order necessary condition opt...

We propose a method to explore invariant measures of dynamical systems. The method is based on numerical tools which directly compute invariant sets using a subdivision technique, and invariant measures by a discretization of the Frobenius-Perron operator. Appropriate visualization tools help to analyze the numerical results and to understand important aspects of the underlying dynamics. This w...

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...

In this paper, we show a new relation between phase transition in Statistical Mechanics and the dimension of space harmonic functions (SHF) for transfer operator. This is accomplished by extending classical Ruelle-Perron-Frobenius theory to realm low regular potentials defined on either finite or infinite (uncountable) alphabets. We also give an example potential having where Perron-Frobenius e...

This paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius-Perron operator for such maps. The method is applied to es...

For a transformation F on a measure space (X,μ), we show that the Perron-Frobenius operator of F can be written by a representation (L2(X,μ), π) of the Cuntz-Krieger algebra OA associated with F when F satisfies some assumption. Especially, when OA is the Cuntz algebra ON and (L2(X,μ), π) in the above is some irreducible representation of ON , then there is an F -invariant measure on X which is...

We study a free boundary problem for the Laplace operator, where we impose a Bernoulli-type boundary condition. We show that there exists a solution to this problem. We use A. Beurling’s technique, by defining two classes of suband supersolutions and a Perron argument. We try to generalize here a previous work of A. Henrot and H. Shahgholian. We extend these results in different directions.