Parameterized family of annular homeomorphisms with pseudo-circle attractors

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 a unique topological object, called the R.H. Bing's pseudo-circle, yet display an interesting boundary dynamics. Namely, in the constructed family of homeomorphisms the outer prime ends rotation number vary continuously with the parameter through the interval [0,1/2]. This, in particular, answers a question from Boroński et al. (2020) [15]. Furthermore, these attractors preserve the induced Lebesgue measure from the circle and have strong measure-theoretic and statistical properties. 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 the pseudo-solenoids. For degree one circle maps, this implies that the generic inverse limit in this context is the pseudo-circle.