We study a diophantine property for translation surfaces, defined in terms of saddle connections and inspired by classical Khinchin condition. We prove that the same dichotomy holds as in Khinchin theorem, then we deduce a sharp estimate on how fast the typical Teichmüller geodesic wanders towards infinity in the moduli space of translation surfaces. Finally we prove some stronger result in genus one.

Let (Ω,A,μ,T) be a measure preserving dynamical system. The speed of convergence in probability in the ergodic theorem for a generic function on Ω is arbitrarily slow.

We consider a hierarchy of notions of largeness for subsets of ℤ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βℤ to establish connections between various notions of largeness and apply those results to the study of the sets $R_{A,B}^{\epsilon }=n\in \mathbb{Z}:\mu \left(A\cap TⁿB\right)>\mu \left(A\right)\mu \left(B\right)-\epsilon$ of times of “fat intersection”. Among other things we show that the sets $R_{A,B}^{\epsilon }$ allow one to distinguish...

On démontre le lemme de Mañé-Conze-Guivarc’h (en classe Lipschitz) pour les systèmes amphidynamiques vérifiant une certaine condition d’hyperbolicité  : la « rectifiabilité ». Diverses applications sont données.

Le théorème classique de Riesz-Raikov assure que, pour tout entier $\theta \>1$ et toute $f$ de $L^1\left(𝕋\right)$, où $𝕋=ℝ/\mathbb{Z}$, les moyennes$$\frac{1}{N}\sum _1^{N}f\left({\theta }^{n}x\right)\text{convergent}\text{vers}{\int }_{𝕋}f\left(t\right)\mathrm{d}t$$pour presque tout point $x$ de $ℝ$. J.Bourgain (cf.Israël Math. Conf. Proc. 1990) a prouvé que la convergence précédente a lieu pour tout réel algébrique $\theta \>1$ et toute $f$ de $L^2\left(𝕋\right)$. Dans cet article nous prouvons que, si $\varphi$ est un endomorphisme de ${ℝ}^p$ algébrique sur $\mathbb{Q}$, dont les valeurs propres sont toutes de module $\>1$, alors pour toute $f$ de $L^2\left({𝕋}^p\right)$, les moyennes $(1/N){\sum }_1^{N}f\left({\varphi }^{n}x\right)$ convergent vers ${\int }_{{𝕋}^p}f\left(t\right)\mathrm{d}t$ pour presque tout point $x$ de ${ℝ}^p$. Nous...

We prove that simple transformations are disjoint from those which are infinitely divisible and embeddable in a flow. This is a reinforcement of a previous result of A. del Junco and M. Lemańczyk [1] who showed that simple transformations are disjoint from Gaussian processes.

Let ${(X_t,Y_t)}_{t\in 𝕋}$ be a discrete or continuous-time Markov process with state space $𝕏\times {ℝ}^d$ where $𝕏$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. ${(X_t,Y_t)}_{t\in 𝕋}$ is assumed to be a Markov additive process. In particular, this implies that the first component ${\left(X_t\right)}_{t\in 𝕋}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process ${\left(Y_t\right)}_{t\in 𝕋}$ is shown to satisfy the...

We prove stable limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from nonintegrable observables over certain piecewise expanding maps. This is done by extending Darling–Kac theory to a suitable family of infinite measure preserving transformations.

We consider the ensemble of curves {γα, N: α∈(0, 1], N∈ℕ} obtained by linearly interpolating the values of the normalized theta sum N−1/2∑n=0N'−1exp(πin2α), 0≤N'<N. We prove the existence of limiting finite-dimensional distributions for such curves as N→∞, when α is distributed according to any probability measure λ, absolutely continuous w.r.t. the Lebesgue measure on [0, 1]. Our Main Theorem generalizes a result by Marklof [Duke Math. J.97 (1999) 127–153] and Jurkat and van Horne [Duke...