Rank one and mixing differentiable flows

We construct, over some minimal translations of the two torus, special flows under a differentiable ceiling function that combine the properties of mixing and rank one

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

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

Deviations of ergodic sums for toral translations II. Boxes

We study the Kronecker sequence $\{n\alpha\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $\alpha$ is uniformly distributed on ${\mathbb T}^d.$ We show that the discrepancy of the number of visits of this sequence to a random box, normalized by $\ln^d N$, converges as $N\to\infty$ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices.Comment: 56 pages. This is a revised and expanded version of the prior submission