  For the family of Double Standard Maps $f_{a,b}=2x+a+\frac{b}{\pi} \sin2\pi x
\quad\pmod{1}$ 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 nonempty 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.
