A classification of minimal sets of torus homeomorphisms

Jäger, Tobias; Kwakkel, Ferry; Passeggi, Alejandro

doi:10.1007/s00209-012-1076-y

Cited by 14 publications

(7 citation statements)

References 23 publications

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“…The classification of compact metric spaces admitting minimal maps is a wellknown open problem in topological dynamics [2,10]. For the state of the art of the problem see [3,8,9,22] and references therein.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Minimal sets of fibre-preserving maps in graph bundles

Kolyada¹,

Snoha²,

Trofimchuk³

2014

Math. Z.

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Topological structure of minimal sets is studied for a dynamical system (E, F) given by a fibre-preserving, in general non-invertible, continuous selfmap F of a graph bundle E. These systems include, as a very particular case, quasiperiodically forced circle homeomorphisms. Let M be a minimal set of F with full projection onto the base space B of the bundle. We show that M is nowhere dense or has nonempty interior depending on whether the set of so called endpoints of M is dense in M or is empty. If M is nowhere dense, we prove that either a typical fibre of M is a Cantor set, or there is a positive integer N such that a typical fibre of M has cardinality N. If M has nonempty interior we prove that there is a positive integer m such that a typical fibre of M, in fact even each fibre of M over a dense open set O ⊆ B, is a disjoint union of m circles. Moreover, we show that each of the fibres of M over B \ O is a union of circles properly containing a disjoint union of m circles. Surprisingly, some of the circles in such "non-typical" fibres of M may intersect. We also give sufficient conditions for M to be a sub-bundle of E.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…To find a full topological characterization of minimal sets on compact, connected 2-manifolds is a very difficult task. Very recently, a classification of minimal sets on 2-torus has been obtained for homeomorphisms [22].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Minimal sets of fibre-preserving maps in graph bundles

Kolyada¹,

Snoha²,

Trofimchuk³

2014

Math. Z.

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“…Kozlowski and the first author [6] proved that if a compact manifold of dimension at least 2 admits a minimal homeomorphism, then it admits a minimal noninvertible map. Minimal sets for surface homeomorphisms were classified in [16] and [26]. In [22] Kolyada, Snoha and Trofimchuk constructed minimal noninvertible maps on T 2 (see also [29]).…”

Section: Introductionmentioning

confidence: 99%

On Rigid Minimal Spaces

2020

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A compact space X is said to be minimal if there exists a map f : X → X such that the forward orbit of any point is dense in X. We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha, and Tywoniuk [J. Dyn. Diff. Eq., 2016] on spaces with cyclic group of homeomorphisms generated by a minimal homeomorphism, and results of the first author, Clark and Oprocha [Adv. Math., 2018] on spaces in which the square of every homeomorphism is a power of the same minimal homeomorphism. We show that the two classes do not coincide, which gives rise to a new class of spaces that admit minimal homeomorphisms, but no minimal maps. We modify the latter class of examples to show for the first time existence of minimal spaces with degenerate homeomorphism groups. Finally, we give a method of constructing decomposable compact and connected spaces with cyclic group of homeomorphisms, generated by a minimal homeomorphism, answering a question in Downarowicz et al.

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“…Kozlowski and the first author [6] proved that if a compact manifold of dimension at least 2 admits a minimal homeomorphism, then it admits a minimal noninvertible map. Minimal sets for surface homeomorphisms were classified in [21] and [27]. In [23] Kolyada, Snoha and Trofimchuk constructed minimal noninvertible maps on T 2 (see also [30]).…”

Section: Introductionmentioning

confidence: 99%

On Multivalent Guiding Functions Method in the Periodic Problem for Random Differential Equations

2019

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A compact space X is said to be minimal if there exists a map f : X → X such that the forward orbit of any point is dense in X. We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha and Tywoniuk (J Dyn Differ Equ, 29:243-257, 2017) on spaces with cyclic group of homeomorphisms generated by a minimal homeomorphism, and results of the first author, Clark and Oprocha (Adv Math, 335:261-275, 2018) on spaces in which the square of every homeomorphism is a power of the same minimal homeomorphism. We show that the two classes do not coincide, which gives rise to a new class of spaces that admit minimal homeomorphisms, but no minimal maps. We modify the latter class of examples to show for the first time existence of minimal spaces with degenerate homeomorphism groups. Finally, we give a method of constructing decomposable compact and connected spaces with cyclic group of homeomorphisms, generated by a minimal homeomorphism, answering a question in Downarowicz et al.

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A classification of minimal sets of torus homeomorphisms

Cited by 14 publications

References 23 publications

Minimal sets of fibre-preserving maps in graph bundles

Minimal sets of fibre-preserving maps in graph bundles

On Rigid Minimal Spaces

On Multivalent Guiding Functions Method in the Periodic Problem for Random Differential Equations

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