Characterizing the Cantor bi-cube in asymptotic categories

Банах, Тарас; Zarichnyi, Ihor

doi:10.4171/ggd/145

Cited by 29 publications

(50 citation statements)

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“…Our first statement shows that, from the asymptotic point of view [23], the scattered subsets of a group can be considered as the counterparts of the scattered subspaces of a topological space. Proposition 1.…”

Section: Resultsmentioning

confidence: 99%

“…Following [23], we say that the set Y of G has no asymptotically isolated balls if Y does not satisfy Propo-…”

Section: Thus Following [24] (Chapter 3) We Can Say Thatmentioning

confidence: 99%

“…By [23], a countable cellular subset Y of G without asymptotically isolated balls is coarsely equivalent to the group ⊕ ! Z 2 .…”

Section: Thus Following [24] (Chapter 3) We Can Say Thatmentioning

confidence: 99%

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Scattered Subsets of Groups

2015

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We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that µ(A) = 0 for each left invariant Banach measure µ on G. It is also shown that every infinite group can be split into @0 scattered subsets.

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Section: Resultsmentioning

confidence: 99%

“…Following [23], we say that the set Y of G has no asymptotically isolated balls if Y does not satisfy Propo-…”

Section: Thus Following [24] (Chapter 3) We Can Say Thatmentioning

confidence: 99%

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Scattered Subsets of Groups

2015

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“…The coarse characterization of the macro-Cantor set M given in [4] implies that the macro-fractal Φ of the multi-function Φ : x → {3x, 3x − 2} is coarsely equivalent to M , so it is legal to call the macro-fractal Φ a macro-Cantor set.…”

Section: Proposition 6 the Fixed Fractal φ Coincides With The Union mentioning

confidence: 99%

“…well-known in the Asymptotic Topology as the asymptotic counterpart of the Cantor set, see [4], [7].…”

Section: A Simple Example Of a Dual Pair Of Micro And Macro Fractalsmentioning

confidence: 99%

Micro and Macro Fractals Generated by Multi-Valued Dynamical Systems

Банах

Novosad

2014

Fractals

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Abstract. Given a multi-valued function Φ : X X on a topological space X we study the properties of its fixed fractal Φ , which is defined as the closure of the orbit Φ ω ( * Φ ) = n∈ω Φ n ( * Φ ) of the set * Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals Φ and Φ −1 for a contracting compact-valued function Φ : X X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.

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