  In this paper we construct a paramaterized family of annular homeomorphisms
with Birkhoff-like rotational attractors that vary continuously with the
parameter, are all homeomorphic to the pseudo-circle, display interesting
boundary dynamics and furthermore preserve the induced Lebesgue measure from
the circle. Namely, in the constructed family of attractors the outer prime
ends rotation number vary continuously with the parameter through the interval
$[0,1/2]$. This, in particular, answers a question from [J. London Math. Soc.
(2) {\bf 102} (2020), 557--579]. To show main results of the paper we first
prove a result of an independent interest, that Lebesgue-measure preserving
circle maps generically satisfy the crookedness condition which implies that
generically the inverse limits of Lebesgue measure-preserving circle maps are
hereditarily indecomposable. For degree one circle maps, this implies that the
generic inverse limit in this context is the R.H. Bing's pseudo-circle.
