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

Let us consider a family of mapsQ _a (x)=ax(1?x) from the unit interval 0,1] to itself, wherea∈0,4] is the parameter. We show that, for any β<2, there exists a subsetE?4 in 0,4] with the properties

1. Leb(4??,4]?E) < ?^β for sufficiently small ?>0,
2. Q _a admits an absolutely continuous BRS measure µ_a whena∈E, and
3. µ_a converges to the measure µ₄ asa tends to 4 on the setE. Also we give some generalization of this results.