4,541 research outputs found

    Rank one and mixing differentiable flows

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    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

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    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

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    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

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    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
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