Hereditarily indecomposable inverse limits of graphs: shadowing, mixing and exactness

Kościelniak, Piotr; Oprocha, Piotr; Tuncali, Murat

doi:10.1090/s0002-9939-2013-11768-5

Cited by 10 publications

(11 citation statements)

References 17 publications

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“…Finally, the inverse limit lim ← − { f, S 1 } is the pseudocircle, since f has degree 1, by the result of Fearnley [17,Theorem 6.3] (for more details see also [19] or [26,Corollary 24] and comments therein) which completes the proof.…”

Section: Auxiliary Lemmasmentioning

confidence: 58%

“…strange attractor with zero topological entropy. Such a construction, if possible, cannot be obtained by an application of the results of [26] (see also [23]) which is an essential ingredient of our approach (see Lemma 3.1).…”

Section: Question Is There a 2-torus Homeomorphism Homotopic To The Imentioning

confidence: 99%

“…The following result from [26] will allow us to produce a circle map f with similar dynamical properties to the map g from Lemma 3.1, but with the pseudocircle as the inverse limit space lim Proof Let g be the map from Lemma 3.1 and letĝ be its lift to the universal cover (R, τ ). Let p and q denote the fixed point and point of period 3, respectively, provided in Lemma 3.1.…”

Section: Auxiliary Lemmasmentioning

confidence: 99%

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Rotational chaos and strange attractors on the 2-torus

Boroński

Oprocha

2014

Math. Z.

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We construct a 2-torus homeomorphism h homotopic to the identity with an attracting R. H. Bing's pseudocircle C such that the rotation set of h|C is not a unique vector. The only known examples of such an attractor were Birkhoff's attractors, arising for dissipative maps with a twist. Our construction relies heavily on the Barge-Martin method for constructing attractors as inverse limits of graphs.

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Section: Auxiliary Lemmasmentioning

confidence: 58%

Section: Question Is There a 2-torus Homeomorphism Homotopic To The Imentioning

confidence: 99%

Section: Auxiliary Lemmasmentioning

confidence: 99%

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Rotational chaos and strange attractors on the 2-torus

Boroński

Oprocha

2014

Math. Z.

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“…In fact a similar approach will be successful, if G is a topological graph with an endpoint p and f is a topologically exact map on G such that f (p) = p. Simply, most of our techniques work locally, so it is really unimportant that G is not an interval. The only essential ingredient is an endpoint which is a fixed point [15] (see also [8]).…”

Section: Mixing But Not Exact Maps and Pseudoarcmentioning

confidence: 99%

“…They provided a method of perturbation (in fact a sequence of perturbations), such that topologically exact map is transformed to a transitive map f such that inverse limit with f as bonding map gives the pseudoarc (it is worth noting, that a few years earlier a map on the pseudoarc with positive topological entropy was constructed by J. Kennedy in [10]). In fact, it can be proved that if f is a transitive map on a topological graph, and inverse limit of this graph with f as a bonding map is hereditarily indecomposable, then f must be mixing [15]. So in fact, all the maps constructed in [12] are topologically mixing.…”

Section: Introductionmentioning

confidence: 99%

Topologically Mixing Maps and the Pseudoarc

Drwięga

Oprocha

2014

Ukr Math J

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TOPOLOGICALLY MIXING MAPS AND THE PSEUDOARC * ТОПОЛОГIЧНО ПЕРЕМIШУЮЧI ВIДОБРАЖЕННЯ ТА ПСЕВДОДУГА It is known that the pseudoarc can be constructed as the inverse limit of the copies of [0, 1] with one bonding map f which is topologically exact. On the other hand, the shift homeomorphism σ f is topologically mixing in this case. Thus, it is natural to ask whether f can be only mixing or must be exact. It has been recently observed that, in the case of some hereditarily indecomposable continua (e.g., pseudocircles) the property of mixing of a bonding map implies its exactness. The main purpose of this article is to show that the indicated kind of forcing of recurrence is not the case for the bonding map defining the pseudoarc. Вiдомо, що псевдодугу можна отримати як обернену границю копiй вiдрiзка [0, 1] з єдиним зв'язуючим вiдображенням f, що є топологiчно точним. З iншого боку, в цьому випадку зсувний гомеоморфiзм σ f є топологiчно перемiшуючим. Таким чином, природно запитати чи може f бути тiльки перемiшуючим, чи воно обов'язково повинно бути точним. Нещодавно було встановлено, що для деяких спадково нерозкладних континуумiв (наприклад, псевдокiл) точнiсть зв'язуючого вiдображення є наслiдком властивостi перемiшування. В данiй статтi показано, що розглянутий тип примусового повернення не реалiзується для зв'язуючого вiдображення, що визначає псевдодугу.

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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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Hereditarily indecomposable inverse limits of graphs: shadowing, mixing and exactness

Cited by 10 publications

References 17 publications

Rotational chaos and strange attractors on the 2-torus

Rotational chaos and strange attractors on the 2-torus

Topologically Mixing Maps and the Pseudoarc

On Rigid Minimal Spaces

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