Expansion properties of double standard maps

Abstract

Abstract For the family of double standard maps $f_{a,b}=2x+a+({b}/{\pi }) \sin 2\pi x \pmod {1}$ we investigate the structure of the space of parameters a when $b=1$ and when $b\in [0,1)$ . In the first case the maps have a critical point, but for a set of parameters $E_1$ of positive Lebesgue measure there is an invariant absolutely continuous measure for $f_{a,1}$ . In the second case there is an open non-empty set $E_b$ of parameters for which the map $f_{a,b}$ is expanding. We show that as $b\nearrow 1$ , the set $E_b$ accumulates on many points of $E_1$ in a regular way from the measure point of view.