1 research outputs found
Multidimensional Gaussian sums arising from distribution of Birkhoff sums in zero entropy dynamical systems
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 of a sequence generated by a skew
product dynamical system on the torus, with zero Lyapounov
exponents. The sequence, taking the values , 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
. We show that when the parameter is rational then all
the moments of the normalized sums , but the second, are
unbounded with respect to n, while for irrational , with bounded
continuous fraction representation, all these moments are finite and bounded
with respect to n.Comment: To be published in J. Phys.