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

Abstract

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 f$f$ generates the pseudo‐arc as inverse limit with f$f$ 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$f$ the background Oxtoby–Ulam measures induced by Lebesgue measure for f$f$ on the interval are physical on the disc and in addition there is a dense set of maps f$f$ 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 false[0,1/2false]$[0,1/2]$. It follows that there are uncountably many dynamically non‐equivalent embeddings of the pseudo‐arc in this family of attractors.

Periodic points and shadowing for generic Lebesgue measure-preserving interval maps

“…on A hence f can not be non-decreasing. Applying the above result, we conclude that In [23, Theorem 1] and [24] the authors proved the following generic property of maps from C λ (M ), which might be the most surprising of the generic properties proven yet: Theorem 2.7.2. There is a dense G δ set T ⊂ C λ (M ) such that if f ∈ T then for every δ > 0 there exists a positive integer n so that f n is (f, δ)-crooked.…”

On conjugacy of natural extensions of one-dimensional maps

“…The following theorem is a restatement of Theorem 5. Note that the interval maps f such that lim ← − ([0, 1], f ) is the pseudo-arc are generic in the closure of the subset of maps of the interval that have a dense set of periodic points [18]. Moreover, all such maps have infinite topological entropy by [35] (see also [11] for a stronger result).…”