Parametrized family of pseudo‐arc attractors: Physical measures and prime end rotations

The main goal of this paper is to study topological and measure‐theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure‐preserving map [Formula: see text] generates the pseudo‐arc as inverse limit with [Formula: see text] as a single bonding map. These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure‐preserving maps [Formula: see text] the background Oxtoby–Ulam measures induced by Lebesgue measure for [Formula: see text] on the interval are physical on the disc and in addition there is a dense set of maps [Formula: see text] defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; that is, the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure‐preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo‐arc attractors with prime ends rotation numbers varying continuously in [Formula: see text]. It follows that there are uncountably many dynamically non‐equivalent embeddings of the pseudo‐arc in this family of attractors.