Expansion properties of double standard maps

Abstract

For the family of Double Standard Maps fa,b = 2x + a + b π sin 2πx (mod 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 nonempty 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.