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Long hitting time, slow decay of correlations and arithmetical properties

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  • 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.

    Mathematics Subject Classification: Primary: 37A25, 37A45; Secondary: 37C45.

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