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...

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 ...

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...

We derive a variational formula for the optimal growth rate of reward in the infinite horizon risk-sensitive control problem for discrete time Markov decision processes with compact metric state and action spaces, extending a formula of Donsker and Varadhan for the Perron-Frobenius eigenvalue of a positive operator. This leads to a concave maximization formulation of the problem of determining ...

In this paper we introduce and study a one-parameter family of piecewise analytic interval maps having the tent map and the Farey map as extrema. Among other things, we construct a Hilbert space of analytic functions left invariant by the Perron-Frobenius operator of all these maps and study the transition between discrete and continuous spectrum when approaching the intermittent situation. AMS...

We study Ruelle–Perron–Frobenius operators for locally expanding and mixing dynamical systems on general compact metric spaces associated with potentials satisfying the Dini condition. In this paper, we give a proof of the Ruelle Theorem on Gibbs measures. It is the first part of our research on the subject. The rate of convergence of powers of the operator will be presented in a forthcoming pa...

In this paper a new quantity for real tensors, the sign-real spectral radius, is defined and investigated. Various characterizations, bounds and some properties are derived. In certain aspects our quantity shows similar behavior to the spectral radius of a nonnegative tensor. In fact, we generalize the Perron Frobenius theorem for nonnegative tensors to the class of real tensors.

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...