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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
; | | Date: |
29 Oct 2004 | | Subject: | Chaotic Dynamics; Mathematical Physics; Dynamical Systems | nlin.CD math-ph math.DS math.MP | | Affiliation: | LPTMC, LPTM), M. Courbage (LPTMC), T.T. Truong (LPTM | | 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 $alpha$. We show that when the parameter $alpha /pi$ 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 $alpha /pi$, with bounded continuous fraction representation, all these moments are finite and bounded with respect to n. | | Source: | arXiv, nlin.CD/0410066 | | Services: | Forum | Review | PDF | Favorites | |
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