Expansion properties of double standard maps
Michael Benedicks, Michał Misiurewicz and Ana Rodrigues
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
increases to 1, the set Eb accumulates on many
points of E1 in a regular way from the measure point
of view.