On the ergodicity of the Weyl sums cocycle

For $\theta \in [0,1]$, we consider the map T_\a: \T^2 \to \T^2 given by $T_\theta(x,y)=(x+\theta,y+2x+\theta)$. The skew product f_\a: \T^2 \times \C \to \T^2 \times \C given by $f_\theta(x,y,z)=(T_\theta(x,y),z+e^{2 \pi i y})$ generates the so called Weyl sums cocycle a_\a(x,n) = \sum_{k=0}^{n-1} e^{2\pi i(k^2\theta+kx)} since the $n^{{\rm th}}$ iterate of f_\a writes as f_\a^n(x,y,z)=(T_\a^n(x,y),z+e^{2\pi iy} a_\a(2x,n)). In this note, we improve the study developed by Forrest in \cite{forrest2,forrest} around the density for x \in \T of the complex sequence {\{a_\a(x,n)\}}_{n\in \N}, by proving the ergodicity of $f_\theta$ for a class of numbers \a that contains a residual set of positive Hausdorff dimension in $[0,1]$. The ergodicity of f_\a implies the existence of a residual set of full Haar measure of x \in \T for which the sequence {\{a_\a(x,n) \}}_{n \in \N} is dense

Non uniform hyperbolicity and elliptic dynamics

We present some constructions that are merely the fruit of combining recent results from two areas of smooth dynamics: nonuniformly hyperbolic systems and elliptic constructions.Comment: 6 pages, 0 figur

An effective version of Katok's horseshoe theorem for conservative C2C^2 surface diffeomorphisms

For area preserving $C^2$ surface diffeomorphisms, we give an explicit finite information condition, on the exponential growth of the number of Bowen's $(n,\delta)-$balls needed to cover a positive proportion of the space, that is sufficient to guarantee positive topological entropy. This can be seen as an effective version of Katok's horseshoe theorem in the conservative setting. We also show that the analogous result is false in dimension larger than $3$

Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary

Let $M$ be an $m$-dimensional differentiable manifold with a nontrivial circle action {\mathcal S}= {\lbrace S_t \rbrace}_{t \in\RR}, S_{t+1}=S_t, preserving a smooth volume $\mu$. For any Liouville number \a we construct a sequence of area-preserving diffeomorphisms $H_n$ such that the sequence H_n\circ S_\a\circ H_n^{-1} converges to a smooth weak mixing diffeomorphism of $M$. The method is a quantitative version of the approximation by conjugations construction introduced in \cite{AK}. For $m=2$ and $M$ equal to the unit disc \DD^2=\{x^2+y^2\leq 1\} or the closed annulus \AAA=\TT\times [0,1] this result proves the following dichotomy: \a \in \RR \setminus\QQ is Diophantine if and only if there is no ergodic diffeomorphism of $M$ whose rotation number on the boundary equals $\alpha$ (on at least one of the boundaries in the case of \AAA). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if \a is Diophantine, then any area preserving diffeomorphism with rotation number \a on the boundary (on at least one of the boundaries in the case of \AAA) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.Comment: To appear in annales de l'EN