MODEL AND ITS MIXING PROPERTIES 1. Setting and Assertions Let f : M \Psi be a C 1+" diffeomorphism of a finite dimensional Riemannian manifold M . In applications we will allow f to have discontinuities or singularities, but these "bad" parts will not appear in the picture we are about to describe. Thus as far as Part I is concerned we may assume that f and f \Gamma1 are defined on all of M . Let d(\Delta; \Delta) denote the distance between points. Riemannian measure on M will be denoted by ; and if W ae M is a submanifold, then W denotes the measure on W induced by the restriction of the Riemannian structure to W . The basic object of interest here consists of a set ae M with a "hyperbolic product structure" and a return map f R from to itself. Precise definitions are given in 1.1 and 1.2; the 4 required properties are listed in (P1)-(P5); and the main results of Part I are stated in 1.4. 1.1. A "horseshoe" with infinitely many branches and variable return times. We be...

The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce sub-optimal function expansions by iteratively choosing the dictionary waveforms which best match the function's structures. Matching pursuits provide a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and using a stochastic differential equation model. These invariant measures define a noise with respect to a given dictionary. ...

. 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 ...

Contents 1 Introduction 1 2 The Computation of Invariant Sets 2 2.1 Brief Review on Invariant Sets . . . . . . . . . . . . . . . . . . 2 2.2 The Computation of Relative Global Attractors . . . . . . . . 3 2.3 Convergence Behavior and Error Estimate . . . . . . . . . . . 6 2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 The Computation of Chain Recurrent Sets . . . . . . . . . . . 9 2.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 11 3 The Computation of Invariant Manifolds 12 3.1 Description of the Method . . . . . . . . . . . . . . . . . . . . 13 3.2 Convergence Behavior and Error Estimate . . . . . . . . . . . 14 3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 15 4 The Computation of SRB-Measures 18 4.1 Brief Review on SRB-Measures and Small Random Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Spectral Approximation for the Per

We consider open sets of maps in a manifold M exhibiting non-uniform expanding behaviour in some domain S ⊂ M. Assuming that there is a forward invariant region containing S where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuously in the L 1-norm with the map. As a main application we show that the open class of maps introduced in [V] fits to this situation, thus proving that the SRB measures constructed in [A] vary continuously with the map. 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.

This paper investigates the decay of correlations in a large class of nonMarkov one-dimensional expanding maps. The method employed is a special version of a general approach recently proposed by the author. Explicit bounds on the rate of decay of correlations are obtained.

We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps with singularities. We allow maps that are discontinuous on some extremely wild sets, the shape of the discontinuities being completely ignored with our approach. Key-Words: Expanding maps with singularities, Invariant measure, Decay of correlations, Perron-Frobenius operator. October 1997 Second version December 1998 CPT-97/P.3552 anonymous ftp : ftp.cpt.univ-mrs.fr web : www.cpt.univ-mrs.fr Unite Propre de Recherche 7061 1 PhyMat, Mathematics Department, University of Toulon. 83957 La Garde, France. Invariant Measure for Multidimensional Expanding Maps 1 Contents 1 Introduction 1 2 Piecewise expanding maps 4 3 Quasi-Holder space 9 4 A Lasota-Yorke type Inequality 12 5 Spectral res...

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

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...