Expansion properties of double standard maps

Abstract

For the family of Double Standard Maps fa;b = 2x + a + b _ sin 2_x (mod 1) we investigate the structure of the space of parameters a when b = 1 and when b 2 [0; 1). In the _rst 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.