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

DOI10.1088/0305-4470/37/49/003zbMATH Open1173.37304arXivnlin/0410066OpenAlexW1993733152MaRDI QIDQ4653024

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Publication date: 28 February 2005

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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 $pm1$, 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 $alpha$. We show that when the parameter $alpha/pi$ is rational then all the moments of the normalized sums $E((S_n/sqrtn{)}^k)$, but the second, are unbounded with respect to n, while for irrational $alpha/pi$, with bounded continuous fraction representation, all these moments are finite and bounded with respect to n.

Full work available at URL: https://arxiv.org/abs/nlin/0410066

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