We estimate the speed of decay of correlations for general nonuniformly expanding dynamical systems, using estimates on the time the system takes to become really expanding. Our method can deal with fast decays, such as exponential or stretched exponential. We prove in particular that the correlations of the Alves-Viana map decay in $O\left({\mathrm{e}}^{-\mathrm{c}\sqrt{\mathrm{n}}}\right)$.

For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which $limin{f}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|=0$ and $limsu{p}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|>0$. We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible definitions...

Nous étudions les flots d’Anosov sur les variétés compactes de dimension 3 pour lesquels les distributions stable et instable faibles sont de classe $C^{\infty }$. Nous classons tous ces flots lorsqu’ils préservent le volume puis nous construisons une famille d’exemples qui ne préservent pas le volume. Nous classons aussi ces flots sous une hypothèse de “pincement”. En application, nous décrivons les déformations des groupes fuchsiens dans le groupe des difféomorphismes du cercle.

According to A. Lasota, a continuous function $f$ from a real compact interval $I$ into itself is called generically chaotic if the set of all points $(x,y)$, for which ${lim\; inf}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|=0$ and ${lim\; sup}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|>0$, is residual in $I\times I$. Being inspired by this definition we say that $f$ is densely chaotic if this set is dense in $I\times I$. A characterization of the generically chaotic functions is known. In the paper the densely chaotic functions are characterized and it is proved that in the class of piecewise monotone maps with finite number of pieces the...

In this paper we prove a Central Limit Theorem for standard kernel estimates of the invariant density of one-dimensional dynamical systems. The two main steps of the proof of this theorem are the following: the study of rate of convergence for the variance of the estimator and a variation on the Lindeberg–Rio method. We also give an extension in the case of weakly dependent sequences in a sense introduced by Doukhan and Louhichi.

In this paper we prove a Central Limit Theorem for standard kernel estimates of the invariant density of one-dimensional dynamical systems. The two main steps of the proof of this theorem are the following: the study of rate of convergence for the variance of the estimator and a variation on the Lindeberg–Rio method. We also give an extension in the case of weakly dependent sequences in a sense introduced by Doukhan and Louhichi.

Let K be the Cantor set. We prove that arbitrarily close to a homeomorphism T: K → K there exists a homeomorphism T̃: K → K such that the ω-limit of every orbit is a periodic orbit. We also prove that arbitrarily close to an endomorphism T: K → K there exists an endomorphism T̃: K → K with every orbit finally periodic.

A new method called C-C-1 method is suggested, which can improve some drawbacks of the original C-C method. Based on the theory of period N, a new quantity S(t) for estimating the delay time window of a chaotic time series is given via direct computing a time-series quantity S(m,N,r,t), from which the delay time window can be found. The optimal delay time window is taken as the first period of the chaotic time series with a local minimum of S(t). Only the first local minimum of the average of a...

Étant donnée une fonction régulière de moyenne nulle sur le tore de dimension $2$, il est facile de voir que ses intégrales ergodiques au-dessus d’un flot de translation “générique”sont bornées. Il y a une dizaine d’années, A. Zorich a observé numériquement une croissance en puissance du temps de ces intégrales ergodiques au-dessus de flots d’hamiltoniens (non-exacts) “génériques”sur des surfaces de genre supérieur ou égal à $2$, et Kontsevich et Zorich ont proposé une explication (conjecturelle) de...

The weak shadowing property is really weaker than the shadowing property. It is proved that every element of the C¹ interior of the set of all diffeomorphisms on a $C^{\infty }$ closed surface having the weak shadowing property satisfies Axiom A and the no-cycle condition (this result does not generalize to higher dimensions), and that the non-wandering set of a diffeomorphism f belonging to the C¹ interior is finite if and only if f is Morse-Smale.

We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup $P_t\phi$ does exist for any bounded $\phi$; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.