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

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