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We are interested in the asymptotic behaviour of the first return time of theorbits of a dynamical system into a small neighbourhood of their startingpoints. We study this quantity in the context of dynamical systems preservingan infinite measure. More precisely, we consider the case of$\mathbb{Z}$-extensions of subshifts of finite type. We also consider a toyprobabilistic model to enlight the strategy of our proofs.

Quantitative recurrence of some dynamical systems with an infinite measure in dimension one