On Rigid Minimal Spaces

Boroński, Jan P.; Činč, Jernej; Foryś-Krawiec, Magdalena

doi:10.1007/s10884-020-09845-4

Cited by 2 publications

(2 citation statements)

References 29 publications

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“…Suppose that the suspension of h embeds in a manifold M. Does every pseudosuspension of h embed in M? Some additional recent results concerning related constructions of 1-dimensional minimal spaces, by various modifications of the suspension method, or from hereditarily indecomposable continua, can be found in [12][13][14]23].…”

Section: Question 117 Does There Exists a Hereditarily Indecomposable Continuum Supporting Minimal Homeomorphism With Infinite Entropy?mentioning

confidence: 99%

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

2021

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We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel–Anosov–Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms with positive finite entropy. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds.We construct an example of a minimal, weakly mixing and uniformly rigid homeomorphism of the pseudo-circle, and by our method we are also able to extend it to other one-dimensional hereditarily indecomposable continua, thereby producing the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension 1. We also show that the examples we construct can be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of connected spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least 2.

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Section: Question 117 Does There Exists a Hereditarily Indecomposable Continuum Supporting Minimal Homeomorphism With Infinite Entropy?mentioning

confidence: 99%

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

2021

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“…Quite recently, in [16] a construction of a family of continua that admit minimal homeomorphisms, but no minimal noninvertible maps was given, resolving Question 1 (see also [11]). However, Conjecture 1 has remained open until now.…”

Section: Introductionmentioning

confidence: 99%

Minimal Non-invertible Maps on the Pseudo-Circle

et al. 2020

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In this article we show that R.H. Bing's pseudo-circle admits a minimal non-invertible map. This resolves a conjecture raised by Bruin, Kolyada and Snoha in the negative. The main tool is a variant of the Denjoy-Rees technique, further developed by Béguin-Crovisier-Le Roux, combined with detailed study of the structure of the pseudo-circle. This is the first example of a planar 1-dimensional space that admits both minimal homeomorphisms and minimal noninvertible maps.

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On Rigid Minimal Spaces

Cited by 2 publications

References 29 publications

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

Minimal Non-invertible Maps on the Pseudo-Circle

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