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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.

*generalized complexity*of an orbit of a dynamical system is defined by the asymptotic behavior of the information that is necessary to describe $n$ steps of the orbit as $n$ increases. This local complexity indicator is also invariant up to topological conjugation and is suited for the study of $0$-entropy dynamical systems. First, we state a criterion to find systems with "non trivial" orbit complexity. Then, we consider also a global indicator of the complexity of the system. This global indicator

*generalizes*the

*topological entropy*, having non trivial values for systems were the number of essentially different orbits increases less than exponentially. Then we prove that if the system is constructive ( if the map can be defined up to any given accuracy by some algorithm) the orbit complexity is everywhere less or equal than the generalized topological entropy. Conversely there are compact non constructive examples where the inequality is reversed, suggesting that the notion of constructive map comes out naturally in this kind of complexity questions.

*weak chaos*, where the information necessary to describe the orbit behavior increases with time more than logarithmically (periodic case) even if less than linearly (positive entropy case). Also, we believe that the above method is useful to classify 0-entropy time series. To support this point of view, we show some theoretical and experimental results in specific cases.

We give general conditions under which the transfer operator is computable on a suitable set. This implies the computability of many "regular enough" invariant measures and among them many physical measures.

On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two examples of computable dynamics, one having a physical measure which is not computable and one for which no invariant measure is computable, showing some subtlety in this kind of problems.

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