Michael Benedicks

On Fourier transforms of functions supported on sets of finite Lebesgue measure

Soit G un groupe abelien localement compact et Ĝ son groupe dual. Soient m et m^ les mesures de Haar sur G et Ĝ. On demontre: soit f∈L 1 (R n ) et f^ sa transformee de Fourier, et soit A={x∈R n ; f(x)¬=0} et B={ζ∈R^ n ; f^(ζ)¬=0}. Alors m(A)<∞ et m^(B)<∞⇒f=0 a.e.[m]. Ici m et m^ denotent la mesure de Lebesgue sur R n resp. R^ n

Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps

We study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.

Almost sure rates of mixing for I.I.D. unimodal maps

It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and 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 and Young [8], and Baladi and Viana [4] 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 stationary measures of i.i.d. itineraries are more difficult to estimate than the “averaged statistics”. Adapting to random systems, on the one hand partitions associated to hyperbolic times due to Alves [1], and on the other a probabilistic coupling method introduced by Young [26] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing.

Absolutely continuous invariant measures for maps with flat tops

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New Developments in the Ergodic Theory of Nonlinear Dynamical Systems

The purpose of this paper is to give a survey of recent results on non-uniformly hyperbolic dynamical systems. The emphasis is on the existence of strange attractors and Sinai-Ruelle-Bowen measures for Henon maps, but we also describe results about statistical properties of such dynamical systems and state some of the open questions in this area.

Corrigendum : Almost sure rates of mixing for I.I.D. unimodal maps

The definition of the return time (p. 117) and the beginning of the proof of Proposition 8.3 of our paper in Vol. 35 of Ann. Scient. Ec. Norm. Sup. (2002) are not correct. We give an amended version which shows that none of the statements are affected. We take this opportunity to correct some other mistakes (without consequences), e.g. in Sublemma 7.2(3) and Lemma 7.10. Published by Editions scientifiques et medicales Elsevier SAS

Decay of random correlation functions for unimodal maps

Since the pioneering results of Jakobson and subsequent work by Benedicks-Carleson and others, it is known that quadratic maps tfa(χ) = a − χ2 admit a unique absolutely continuous invariant measure for a positive measure set of parameters a. For topologically mixing tfa, Young and Keller-Nowicki independently proved exponential decay of correlation functions for this a.c.i.m. and smooth observables. We consider random compositions of small perturbations tf +ωt, with tf = tfa or another unimodal map satisfying certain nonuniform hyperbolicity axioms, and ωt chosen independently and identically in [−ϵ, ϵ]. Baladi-Viana showed exponential mixing of the associated Markov chain, i.e., averaging over all random itineraries. We obtain stretched exponential bounds for the random correlation functions of Lipschitz observables for the sample measure μωof almost every itinerary.