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Multidimensional Gaussian sums arising from distribution of Birkhoff sums in zero entropy dynamical systems

M. Bernardo·M. Courbage·T. T. Truong
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Abstract

A duality formula, of the Hardy and Littlewood type for multidimensional Gaussian sums, is proved in order to estimate the asymptotic long time behavior of distribution of Birkhoff sums $S_n$ of a sequence generated by a skew product dynamical system on the $\mathbb{T}^2$ torus, with zero Lyapounov exponents. The sequence, taking the values $\pm 1$, is pairwise independent (but not independent) ergodic sequence with infinite range dependence. The model corresponds to the motion of a particle on an infinite cylinder, hopping backward and forward along its axis, with a transversal acceleration parameter $α$. We show that when the parameter $α/π$ is rational then all the moments of the normalized sums $E((S_n/\sqrt{n})^k)$, but the second, are unbounded with respect to n, while for irrational $α/π$, with bounded continuous fraction representation, all these moments are finite and bounded with respect to n.

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M. Bernardo, M. Courbage, T. T. Truong. 2004-10-29. Multidimensional Gaussian sums arising from distribution of Birkhoff sums in zero entropy dynamical systems. https://doi.org/10.1088/0305-4470%2F37%2F49%2F003

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