On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps

Let us consider a family of maps Q_a(x) = ax(1 - x) from the unit interval [0, 1] to itself, where a ∈ [0, 4] is the parameter. We show that, for any β < 2, there exists a subset E ∋ 4 in [0, 4] with the properties (1) Leb([4 - ε, 4] - E) < ε^β for sufficiently small ε > 0, (2) Q_a admits an absolutely continuous BRS measure μ_a when a ∈ E, and (3) μ_a converges to the measure μ₄ as a tends to 4 on the set E. Also we give some generalization of this results.