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.

We consider measure-preserving diffeomorphisms of the torus with zero entropy. We prove that every ergodic $C^1$-diffeomorphism with linear growth of the derivative is algebraically conjugate to a skew product of an irrational rotation on the circle and a circle $C^1$-cocycle. We also show that for no positive β ≠ 1 does there exist an ergodic $C^2$-diffeomorphism whose derivative has polynomial growth with degree β.