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

Abstract

Abstract. We consider the quadratic family of maps given by f a (x) = 1 − ax 2 on I = [−1, 1], for a positive Lebesgue measure set of parameters close to a = 2-the BenedicksCarleson parameters, on which there is exponential growth of the derivative of the critical point and an absolutely continuous SRB invariant measure. We show that the volume of the set of points of I that 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.

Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems

“…That is, the map a → ρ a = dν a /dm from Q b,c → L 1 is continuous. (See also [15,16] for the same result restricted to the Benedicks-Carleson parameters [10]. )…”

Almost sure mixing rates for non-uniformly expanding maps

“…We stress that this does not imply that 0 belongs to the support of θ ǫ . We consider Ω = Ω ǫ = T Z , P = P ǫ = θ Z ǫ and for t ∈ T we set Φ(t) = f t (x) = f (x) + t. We assume that f is a transitive non-uniformly expanding and non-uniformly expanding (with slow recurrence to the critical set C = {0}) and notice that it holds for f a with Benedicks-Carleson parameter a; see [25]. We will see that we are in conditions to apply Theorem B to get the following result.…”

Semicontinuity of entropy, existence of equilibrium states and continuity of physical measures

“…In a recent work [21] the orbits of these maps where shown to have slow approximation to the non-flat critical set C = {0}.…”