Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps

We consider the quadratic family of maps given by f_a(x) = 1 − ax² on I = [−1, 1], for the Benedicks–Carleson parameters. On this positive Lebesgue measure set of parameters close to a = 2, f_a presents an exponential growth of the derivative along the orbit of the critical point and has an absolutely continuous Sinai–Ruelle–Bowen (SRB) invariant measure. We show that the volume of the set of points of I which, at a given time, fail to present an exponential growth of the derivative, decays exponentially as time passes. We also show that the set of points of I that are not slowly recurrent to the critical set decays sub-exponentially. As a consequence, we obtain continuous variation of the SRB measures and associated metric entropies with the parameter on the referred set. For this purpose, we elaborate on the Benedicks–Carleson techniques in the phase space setting.