4,541 research outputs found

### 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

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