liminf_{r → 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 limsup_{r → 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.
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