# American Institute of Mathematical Sciences

February  2010, 27(1): 185-204. doi: 10.3934/dcds.2010.27.185

## Long hitting time, slow decay of correlations and arithmetical properties

 1 Dipartimento di Matematica Applicata, Via Buonarroti 1, Pisa, 56100, Italy 2 Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, 56100, Italy

Received  March 2009 Revised  July 2009 Published  February 2010

Let $\tau _r(x,x_0)$ be the time needed for a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We construct a class of translations on the two torus having particular arithmetic properties (Liouville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for typical $x,x_0$

liminfr → 0 $\frac{\log \tau _r(x,x_0)}{-\log r} = \infty.$

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 limsupr → 0 of the above ratio of logarithms (which is also called the upper hitting time indicator) is bounded from above by 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.

Citation: Stefano Galatolo, Pietro Peterlongo. Long hitting time, slow decay of correlations and arithmetical properties. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 185-204. doi: 10.3934/dcds.2010.27.185
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