We estimate the speed of decay of correlations for general nonuniformly expanding dynamical systems, using estimates on the time the system takes to become really expanding. Our method can deal with fast decays, such as exponential or stretched exponential. We prove in particular that the correlations of the Alves-Viana map decay in $O\left({\mathrm{e}}^{-\mathrm{c}\sqrt{\mathrm{n}}}\right)$.

For the real quadratic map $P_a\left(x\right)=x^2+a$ and a given $ϵ\>0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_a^n{|}_J$ univalent, with bounded distortion and $B(0,ϵ)\subseteq P_a^n\left(J\right)$ for some $n\in \mathbb{N}$. The $ϵ$-weakly expanding set is the set of points which do not have good expansion properties. Let $\alpha$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[\alpha ,-\alpha ]$. We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff dimension of...