4,541 research outputs found

    Rank one and mixing differentiable flows

    Full text link
    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

    Full text link
    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

    Full text link
    For θ[0,1]\theta \in [0,1], we consider the map T_\a: \T^2 \to \T^2 given by Tθ(x,y)=(x+θ,y+2x+θ)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 fθ(x,y,z)=(Tθ(x,y),z+e2πiy)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 nthn^{{\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 fθf_\theta for a class of numbers \a that contains a residual set of positive Hausdorff dimension in [0,1][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

    Get PDF
    We study the Kronecker sequence {nα}nN\{n\alpha\}_{n\leq N} on the torus Td{\mathbb T}^d when α\alpha is uniformly distributed on Td.{\mathbb T}^d. We show that the discrepancy of the number of visits of this sequence to a random box, normalized by lndN\ln^d N, converges as NN\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+1d+1 dimensional lattices.Comment: 56 pages. This is a revised and expanded version of the prior submission