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Skew products, quantitative recurrence, shrinking targets and decay of correlations

We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recurrence (also with respect to given observables) and hitting time scale behavior depend on the arithmetical properties of the extension. By this we show that those systems have a polynomial decay of correlations with respect to $C^{r}$ observables, and give estimations for its exponent, which depend on $r$ and on the arithmetical properties of the system. We also show examples of systems of this kind having not the shrinking target property, and having a trivial limit distribution of return time statistics.