Viviane Baladi

Zeta functions and transfer operators for piecewise monotone transformations

Given a piecewise monotone transformationT of the interval and a piecewise continuous complex weight functiong of bounded variation, we prove that the Ruelle zeta function ζ(z) of (T, g) extends meromorphically to {∣z∣<θ-1} (where θ=lim ∥g°Tn-1...g°Tg∥∞1/n) and thatz is a pole of ζ if and only ifz−1 is an eigenvalue of the corresponding transfer operator L. We do not assume that L leaves a reference measure invariant.

Periodic orbits and dynamical spectra (Survey)

Basic results in the rigorous theory of weighted dynamical zeta functions or dynamically defined generalized Fredholm determinants are presented. Analytic properties of the zeta functions or determinants are related to statistical properties of the dynamics via spectral properties of dynamical transfer operators, acting on Banach spaces of observables.

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.

Abnormal escape rates from nonuniformly hyperbolic sets

Consider a C 1+ diieomorphism f having a uniformly hyperbolic compact invariant set , maximal invariant in some small neighbourhood of itself. The asymp-totic exponential rate of escape from any small enough neighbourhood of is given by the topological pressure of ? log jJ u fj on (Bowen{Ruelle 1975]). It has been conjectured (Eckmann{Ruelle 1985]) that this property, formulated in terms of escape from the support of a (generalized SRB) measure, using its entropy and positive Lyapunov exponents , holds more generally. We present a simple C 1 two-dimensional counterexample, constructed by a surgery using a Bowen-type hyperbolic saddle attractor as the basic plug.

Lyapunov exponents for non-classical multidimensional continued fraction algorithms

We introduce a simple geometrical two-dimensional continued fraction algorithm inspired from dynamical renormalization. We prove that the algorithm is weakly convergent, and that the associated transformation admits an ergodic absolutely continuous invariant probability measure. Following Kosygin and Baldwin, its Lyapunov exponents are related to the approximation exponents which measure the diophantine quality of the continued fraction. The Lyapunov exponents for our algorithm, and related ones also introduced in this article, are studied numerically.

A Brief Introduction to Dynamical Zeta Functions

We shall take the point of view that dynamical zeta functions are useful objects to describe the spectrum of transfer operators. To define a transfer operator, we use two ingredients: a map f : X → X of a topological or metric space X to itself (the dynamical system), with the property that f -1 (x) is an at most countable set for each x ∈ X; a weight g : X → C.

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.

Ergodicity and Mixing

The notion of ergodicity was introduced by L. Boltzmann in connection with Foundations of Statistical Mechanics. Now its role for Statistical Mechanics is not so much clear but it is very important for the theory of dynamical systems and deterministic chaos.