S-limit shadowing is generic for continuous Lebesgue measure preserving circle maps

Bobok, Jozef; Činč, Jernej; Oprocha, Piotr; Troubetzkoy, Serge

doi:10.48550/arxiv.2104.03999

Cited by 1 publication

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“…For preliminary results concerning crookedness we adjust in Section 2 techniques developed by Minc and Transue [41] and combine them with a special window perturbations that were first introduced in [18] and subsequently used in [16,17]. Of central importance in proving Theorem 1.1 is Lemma 2.20, where we show that the Lebesgue measure-preserving perturbations we construct satisfy certain requirements from [41].…”

Section: Insight Into the Proof And The Outline Of The Papermentioning

confidence: 99%

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

Činč

Oprocha

2022

Proceedings of London Math Soc

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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.

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