Lower bounds for the decay of correlations in non-uniformly expanding maps

We give conditions under which non-uniformly expanding maps exhibit lower bounds of polynomial type for the decay of correlations and for a large class of observables. We show that if the Lasota–Yorke-type inequality for the transfer operator of a first return map is satisfied in a Banach space ${\mathcal{B}}$, and the absolutely continuous invariant measure obtained is weak mixing, in terms of aperiodicity, then, under some renewal condition, the maps have polynomial decay of correlations for observables in ${\mathcal{B}}.$ We also provide some general conditions that give aperiodicity for expanding maps in higher dimensional spaces. As applications, we obtain lower bounds for piecewise expanding maps with an indifferent fixed point and for which we also allow non-Markov structure and unbounded distortion. The observables are functions that have bounded variation or satisfy quasi-Hölder conditions and have their support bounded away from the neutral fixed points.

Stochastic Perturbations and Invariant Measures of Position Dependent Random Maps via Fourier Approximations

Let T = {τ1(x), τ2(x),…, τK(x); p1(x), p2(x),…, pK(x)} be a position dependent random map which possesses a unique absolutely continuous invariant measure [Formula: see text] with probability density function [Formula: see text]. We consider a family {TN}N≥1 of stochastic perturbations TN of the random map T. Each TN is a Markov process with the transition density [Formula: see text], where qN(x, ⋅) is a doubly stochastic periodic and separable kernel. Using Fourier approximation, we construct a finite dimensional approximation PN to a perturbed Perron–Frobenius operator. Let [Formula: see text] be a fixed point of PN. We show that [Formula: see text] converges in L1 to [Formula: see text].

Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps

We consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension.

W-LIKE MAPS WITH VARIOUS INSTABILITIES OF ACIM'S

This paper generalizes the results of [Li et al., 2011] and then provides an interesting example. We construct a family of W-like maps {Wa} with a turning fixed point having slope s1 on one side and –s2 on the other. Each Wa has an absolutely continuous invariant measure μa. Depending on whether [Formula: see text] is larger, equal or smaller than 1, we show that the limit of μa is a singular measure, a combination of singular and absolutely continuous measure or an absolutely continuous measure, respectively. It is known that the invariant density of a single piecewise expanding map has a positive lower bound on its support. In Sec. 4 we give an example showing that in general, for a family of piecewise expanding maps with slopes larger than 2 in modulus and converging to a piecewise expanding map, their invariant densities do not necessarily have a uniform positive lower bound on the support.

INVARIANT MEASURES FOR RANDOM MAPS VIA INTERPOLATION

Let T = {τ1, τ2, …, τK; p1, p2, …, pK} be a position dependent random map on [0, 1], where {τ1, τ2, …, τK} is a collection of nonsingular maps on [0, 1] into [0, 1] and {p1, p2, …, pK} is a collection of position dependent probabilities on [0, 1]. We assume that the random map T has a unique absolutely continuous invariant measure μ with density f*. Based on interpolation, a piecewise linear approximation method for f* is developed and a proof of convergence of the piecewise linear method is presented. A numerical example for a position dependent random map is presented.

Optimal transport and dynamics of expanding circle maps acting on measures

In this paper we compute the derivative of the action on probability measures of an expanding circle map at its absolutely continuous invariant measure. The derivative is defined using optimal transport: we use the rigorous framework set up by Gigli to endow the space of measures with a kind of differential structure. It turns out that 1 is an eigenvalue of infinite multiplicity of this derivative, and we deduce that the absolutely continuous invariant measure can be deformed in many ways into atomless, nearly invariant measures. We also show that the action of standard self-covering maps on measures has positive metric mean dimension.