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We study partially hyperbolic sets $K$ on a Riemannian manifold $M$ whosetangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which thecenter-unstable direction $E^{cu}$ is non-uniformly expanding on some localunstable disk. We prove that the (stretched) exponential decay of recurrencetimes for an induced scheme can be deduced under the assumption of (stretched)exponential decay of the time that typical points need to achieve some uniformexpanding in the center-unstable direction. This extends a result by Alves andPinheiro to the (stretched) exponential case. As an application of our mainresult we obtain (stretched) exponential decay of correlations andexponentially large deviations for a class of partially hyperbolicdiffeomorphisms introduced by Alves, Bonatti and Viana.

Gibbs-Markov-Young structures with (stretched) exponential tail for partially hyperbolic attractors