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

Long hitting time, slow decay of correlations and arithmetical properties


Dipartimento di Matematica Applicata, Via Buonarroti 1, Pisa, 56100, Italy


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

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