Parametrized family of pseudo-arc attractors: physical measures and prime end rotations

TL;DR

This paper investigates a family of strange planar attractors, demonstrating their topological and measure-theoretic properties, including the existence of physical measures, statistical stability, and continuous variation of prime end rotation numbers.

Contribution

It constructs a parametrized family of pseudo-arc attractors with varying prime end rotation numbers and establishes their measure-theoretic stability and topological properties.

Findings

Pseudo-arc appears as inverse limit of Lebesgue measure preserving maps.

Attractors vary continuously with parameters in Hausdorff distance.

Family of physical measures is statistically stable and varies continuously.

Abstract

The main goal of this paper is to study topological and measure-theoretic properties of an intriguing family of strange planar attractors. Building towards these results, we first show that any generic Lebesgue measure preserving map $f$ generates the pseudo-arc as inverse limit with $f$ 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 $f$ the background Oxtoby-Ulam measures induced by Lebesgue measure for $f$ on the interval are physical on the disc and in addition there is a dense set of maps $f$ defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; i.e. 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 $[0,1/2]$. It follows that there are uncountably many dynamically non-equivalent embeddings of the pseudo-arc in this family of attractors.