We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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.
For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random 
$\beta $
-transformations and random Lüroth maps with a hole.
