We consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing. We do this by proving some simple lemmas about the robustness of the spectra of certain operators. These abstract results are then applied to the Perron-Frobenius operators of the models in question. Introduction Let f : M → M be a dynamical...

We introduce the block numerical range Wn(L ) of an operator function L with respect to a decomposition H = H1⊕ . . .⊕Hn of the underlying Hilbert space. Our main results include the spectral inclusion property and estimates of the norm of the resolvent for analytic L . They generalise, and improve, the corresponding results for the numerical range (which is the case n = 1) since the block nume...

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...

I present numerical methods for the computation of stable and unstable manifolds in autonomous dynamical systems. Through differentiation of the Lyapunov-Perron operator in [1], we find that the stable and unstable manifolds are boundary value problems on the original set of differential equation. This allows us to create a forward-backward approach for manifold computation, where we iterativel...

We develop a new collection of tools aimed at studying stochastically perturbed dynamical systems. Specifically, in the setting of bi-stability, that is a two-attractor system, it has previously been numerically observed that a small noise volume is sufficient to destroy would be zero-noise case barriers in the phase space (pseudo-barriers), thus creating a pre-heteroclinic tangency chaos-like ...

We consider the classical evolution of a particle on a graph by using a time-continuous Frobenius-Perron operator that generalizes previous propositions. In this way, the relaxation rates as well as the chaotic properties can be defined for the time-continuous classical dynamics on graphs. These properties are given as the zeros of some periodic-orbit zeta functions. We consider in detail the c...