Invariant measures of higher dimensional transformations are hard to calculate. We present new results on the estimation of absolutely continous invariant measures of mixing transformations , including a new method of proof of Ulam's conjecture. The method involves constructing nite matrix approximations to the Perron-Frobenius operator from increasingly ner partitions of the state space X. We ...

For all hyperbolic polynomials we proved in [11] a Lipschitz estimate of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics (with real Julia set). Here we prove that for all sufficiently hyperbolic polynomials this estimate becomes exponentially better when the dim...

We study the properties of the Google matrix generated by a coarse-grained Perron-Frobenius operator of the Chirikov typical map with dissipation. The finite-size matrix approximant of this operator is constructed by the Ulam method. This method applied to the simple dynamical model generates directed Ulam networks with approximate scale-free scaling and characteristics being in certain feature...

This article is about the data-driven computation of optimal control for a class affine deterministic nonlinear systems. We assume that dynamical system model not available, and only information dynamics available in form time-series data. provide convex formulation problem (OCP) system. The relies on duality result system’s stability theory involving density function Perron–Frobenius operator....

We report the numerical observation of scarring, that is enhancement probability density around unstable periodic orbits a chaotic system, in eigenfunctions classical Perron-Frobenius operator noisy Anosov ("cat") maps, as well Bunimovich stadium. A parallel drawn between and quantum scars, based on unitarity or non-unitarity respective propagators. For uniformly hyperbolic systems such cat map...

The purpose of this paper is to present a unified Perron-Frobenius Theory for nonnegative, for real not necessarily nonnegative and for general complex matrices. The sign-real spectral radius was introduced for general real matrices. This quantity was shown to share certain properties with the Perron root of nonnegative matrices. In this paper we introduce the sign-complex spectral radius. Agai...

We introduce a relative notion of the ’big images and preimages’-property for random topological Markov chains. This then implies that a relative version of the Ruelle-Perron-Frobenius theorem holds with respect to summable and locally Hölder continuous potentials.

We use the natural invariant density of map and Perron–Frobenius operator to analytically evaluate statistical properties for chaotic intermittency. This study can be understood as an improvement previous ones because it does not introduce assumptions about reinjection probability function in laminar interval or at pre-reinjection points. To validate new theoretical equations, we a symmetric no...