Expansion properties of double standard maps

This is the authors' abstract. We don't add key points for this paper.

For the family of double standard maps fa,b=2x+a+(b/π ) sin 2π x 1 we investigate the structure of the space of parameters a when $b=1$ and when b∈ [0,1) . In the first case the maps have a critical point, but for a set of parameters E₁ of positive Lebesgue measure there is an invariant absolutely continuous measure for fa,1 . In the second case there is an open non-empty set Eb of parameters for which the map fa,b is expanding. We show that as b 1 , the set Eb accumulates on many points of E₁ in a regular way from the measure point of view.