. We consider small random perturbations of a large class of nonuniformly hyperbolic unimodal maps and prove stochastic stability in the strong sense (L 1 -convergence of invariant densities) and uniform bounds for the exponential rate of decay of correlations. Our method is based on an analysis of the spectrum of a modified Perron-Frobenius operator for a tower extension of the Markov chain. 1. Introduction Let I ae R be a compact interval and f : I ! I be a smooth unimodal map with f(I) ae int (I). The prototype we have in mind are the quadratic maps f(x) = \Gammax 2 + a but our arguments and conclusions hold in the general context of maps with negative Schwarzian derivative and nondegenerate critical point. Let c 2 I be the critical point of f and c k = f k (c) for k 0. Throughout this paper we assume that (A1) jf k (c) \Gamma cj e \Gammaffk for all k H 0 , (A2) j(f k ) 0 (c 1 )j k c for all k H 0 , (A3) f is topologically mixing on the interval bounded by c ...

We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov di#eomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem. 1.

We prove stability of the isolated eigenvalues of transfer operators satisfying a Lasota-Yorke type inequality under a broad class of random and nonrandom perturbations including Ulamtype discretizations. The results are formulated in an abstract framework.

For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here "eigenvalue" means eigenvalue of the corresponding Perron-Frobenius operator acting on the space of functions of bounded variation. ) This result applies e.g. to the approximation of the system by a finite state Markov chain and generalizes Ulam's conjecture about the approximation of the SBR invariant measure of such a map. We provide several simple examples showing that for maps with periodic turning points and for general multidimensional smooth hyperbolic maps isolated eigenvalues are typically unstable under random perturbations. Our main tool in the 1D case is a special technique for "interchanging" the map and the perturbation, developed in our previous paper [6], combined with a compactness argument. 1 Introduction We discuss stochastic stability in the fo...

We describe a computational method of approximating the "physical" or Sinai-Bowen-Ruelle measure of an Anosov system in two dimensions. The approximation may either be viewed as a fixed point of an approximate Perron-Frobenius operator or as an invariant measure of a randomly perturbed system. Keywords: invariant measure, Perron-Frobenius operator, small random perturbation. 1991 Mathematics Subject Classification. Primary 58F11, 58F30; Secondary 28D05, 41A65. 1 Introduction The existence and computation of important invariant measures of deterministic dynamical systems are still major concerns in ergodic theory. In this note, we do not address the problem of existence, but provide a small step in the computation of important measures when they are known to exist. In one dimension, absolutely continuous invariant measures are considered to be important from a computational point of view because it is absolutely continuous measures that show up on computer simulations for most starting...

. We consider random compositions F oe n ! ffi \Delta \Delta \Delta ffi F! of C k expanding maps F! which are C k -close to a given C k expanding map (k ? 1) and not necessarily i.i.d. We study the random correlation functions C! (n) associated to the unique absolutely continuous stationary measures F!! = oe! and smooth test functions. We show C k\Gamma1 stability of the densities of the measures ! , and good uniform bounds on the exponential rate of decay of random correlations as the smooth error level goes to zero. To do this, we let the associated random transfer operators LF! act on suitable cones of positive functions endowed with a Hilbert projective metric. 1. Introduction When studying small random perturbations of a given expanding dynamical system f : X ! X, i.e., compositions F oe n ! ffi \Delta \Delta \Delta ffi F ! with each random variable F ! "close" to f (see Section 2 for precise definitions), one approach is to consider the Markov chain with transiti...

. It has been known since the pioneering work of Jakobson and subsequent work by Benedicks-Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and Keller-Nowicki proved exponential decay of its correlation functions. Benedicks-Young [BeY] and Baladi-Viana [BV] studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical properties of the sample measures of i.i.d. itineraries are more dicult to estimate than the \averaged statistics." Adapting to random systems, on the one hand the notion of hyperbolic times due to Alves [A], and on the other a probabilistic coupling method introduced by Young [Yo2] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing. 1. Introduction An important class of discrete-time deterministic dynamical systems (given by a transformation f on a Rieman...

I explore the concrete applicability of recent theoretical results to the rigorous computation of relevant statistical properties of a simple class of dynamical systems: piecewise expanding maps 1 Introduction The aim of the present paper is to investigate the possibility of answering questions of the type: ffl Given a piecewise expanding map is it possible to decide if it is ergodic or mixing? ffl Is it possible to determine with arbitrary precision its absolutely continuous invariant measure? ffl If the map is mixing, is it possible to compute the exact rate of decay of correlations for a given function? Of course, the literature contains many papers in which some of these question are discussed either theoretically (especially, but not exclusively, as far as the invariant density is concerned) or numerically (e.g. [3], [4, 5, 6, 7], [8, 9], [14], [15], [18, 19], [21, 22], [23, 24, 25], [27, 28, 29, 30, 31, 32, 33], [34, 35], [38], [39, 40], [48], [49], [52], [55], [62], [66]). N...

We consider weakly coupled analytic expanding circle maps on the lattice Z d (for d ≥ 1), with small coupling strength ɛ and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a Fréchet space on which the operator associated to the full system has a simple eigenvalue at 1 (corresponding to the SRB measure µɛ previously obtained by Bricmont–Kupiainen [BK1]) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For d = 1 we also construct Banach spaces of densities with respect to µɛ on which perturbation theory, applied to the difference of fixed high iterates of the normalised coupled and uncoupled transfer operators, yields localisation of the full spectrum of the coupled operator (i.e., the first spectral gap and beyond). As a side-effect, we show that the whole spectra of the truncated coupled transfer operators (on bounded analytic functions) are O(ɛ)-close to the truncated uncoupled spectra, uniformly in the spatial size. Our method uses polymer expansions and also gives the exponential decay of time-correlations for a larger class of observables than those considered in [BK1].

n=0 n X(y)ρ0(y) ∂ ∂y ϕ(fn (y)) dy associated to the perturbation ft = f + tX of a piecewise expanding interval map f, and to an observable ϕ. The analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions on f, X, and ϕ which guarantee that Ψ(z) is holomorphic in a disc of larger than one. Although R Ψ(1) is the formal derivative (at t = 0) of the average R(t) = ϕρt dx of ϕ with respect to the SRB measure of ft, we present examples of f, X, and ϕ satisfying our conditions so that R(t) is not Lipschitz at 0. that the set {x ∈ M | limn→ ∞ 1 n 1. Introduction and