Kočergin introduced in 1975 a class of smooth flows on the two torus that are mixing. When these flows have one fixed point, they can be viewed as special flows over an irrational rotation of the circle, with a ceiling function having a power-like singularity. Under a Diophantine condition on the rotation’s angle, we prove that the special flows actually have a $t^{-\eta }$-speed of mixing, for some $\eta \>0$.

A random map is a discrete-time dynamical system in which one of a number of transformations is randomly selected and applied on each iteration of the process. We study random maps with position dependent probabilities on the interval and on a bounded domain of ℝⁿ. Sufficient conditions for the existence of an absolutely continuous invariant measure for a random map with position dependent probabilities on the interval and on a bounded domain of ℝⁿ are the main results.

E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $\left|\right|Tⁿ\left|\right|\sim n^{1/4}$). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $li{m}_{n\to \infty }\left|\right|Tⁿ\left|\right|/n^{1-\gamma }=\infty .IntheclassofpositiveoperatorstheseexamplesarethebestpossibleinthesensethatforeverysuchoperatorTthereexistsa\gamma ₀>0suchthat$lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0$.$A class of numerical sequences αₙ, intimately related to the...

We show that a certain type of quasifinite, conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic, measure preserving transformation which is not quasifinite; and consider distribution asymptotics of information showing that e.g. for Boole's transformation, information is asymptotically mod-normal with normalization ∝ √n. Lastly, we show that certain ergodic, probability preserving...

We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) $li{m}_{n\to \infty }{n}^{-1}log{\sum }_{j=0}^{\tau ₙ\left(x\right)}\mu \left(\alpha ⁿ\left(T^j\left(x\right)\right)\right)$, where $\alpha ⁿ\left(T^j\left(x\right)\right)$ is the element of the partition containing $T^j\left(x\right)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

Récemment, B. Green et T. Tao ont montré que : l’ensemble des nombres premiers contient des progressions arithmétiques de toutes longueurs répondant ainsi à une question ancienne à la formulation particulièrement simple. La démonstration n’utilise aucune des méthodes “transcendantes” ni aucun des grands théorèmes de la théorie analytique des nombres. Elle est écrite dans un esprit proche de celui de la théorie ergodique, en particulier de celui de la preuve par Furstenberg du théorème de Szemerédi,...

This work is devoted to the concept of statistical solution of the Navier-Stokes equations, proposed as a rigorous mathematical object to address the fundamental concept of ensemble average used in the study of the conventional theory of fully developed turbulence. Two types of statistical solutions have been proposed in the 1970’s, one by Foias and Prodi and the other one by Vishik and Fursikov. In this article, a new, intermediate type of statistical solution is introduced and studied. This solution...

In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if $f\in L^p$, $p$ large enough, is a Wiener-Wintner function then, for all $\gamma \in (1+\frac{1}{2p}-\frac{\beta }{2},1]$, there exists a set $X_f$ of full measure for which the series ${\sum }_{n=1}^{\infty }\frac{f\left(T^{n}x\right)e^{2\pi inϵ}}{n^{\gamma }}$ converges uniformly with respect to $ϵ$.

Nous étudions certaines propriétés combinatoires, ergodiques et arithmétiques du point fixe de la substitution de Tribonacci (introduite par G. Rauzy) et de la rotation du tore ${𝕋}^2$ qui lui est associée. Nous établissons une généralisation géométrique du théorème des trois distances et donnons une formule explicite pour la fonction de récurrence du point fixe. Nous donnons des propriétés d’approximation diophantienne du vecteur de la rotation de ${𝕋}^2$ : nous montrons, que pour une norme adaptée, la suite...

A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space ${}_{\phi }$ has pure discrete spectrum, we describe the collection ${}_{\phi }^P$ of pairs of proximal tilings in ${}_{\phi }$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then ${}_{\phi }$ and ${}_{\psi }$ are homeomorphic if and...

Entropy-expanding transformations define a class of smooth dynamics generalizing interval maps with positive entropy and expanding maps. In this work, we build a symbolic representation of those dynamics in terms of puzzles (in Yoccoz’s sense), thus avoiding a connectedness condition, hard to satisfy in higher dimensions. Those puzzles are controled by a «constraint entropy» bounded by the hypersurface entropy of the aforementioned transformations.The analysis of those puzzles rests on a «stably...