ar X iv : 0 80 1 . 31 09 v 4 [ m at h . D S ] 2 3 M ar 2 00 9 LONG HITTING TIME , SLOW DECAY OF CORRELATIONS AND ARITHMETICAL PROPERTIES

Let τr(x, x 0) be the time needed for a point x to enter for the first time in a ball Br(x 0) centered in x 0 , with small radius r. We construct a class of translations on the two torus having particular arithmetic properties (Liou-ville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for generic x, x 0 lim inf r→0 log τr(x, x 0) − log r = ∞. By considering a suitable reparametrization of the flow generated by a suspension of this translation, using a previous construction by Fayad, we show the existence of a mixing system on three torus having the same properties. The speed of mixing of this example must be subpolynomial, because we also show that: in a system having polynomial decay of correlations the above ratio of logarithms (which is also called the lower hitting time indicator) is bounded (it is a function of the local dimension and the speed of correlation decay). More generally, this shows that reparametrizations of torus translations having a Liouville component cannot be polynomially mixing.