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Composants of the horseshoe

100%
Bandt C.
Fundamenta Mathematicae
|
1994
|
tom 144
|
nr 3
231-241
EN
The horseshoe or bucket handle continuum, defined as the inverse limit of the tent map, is one of the standard examples in continua theory as well as in dynamical systems. It is not arcwise connected. Its arcwise components coincide with composants, and with unstable manifolds in the dynamical setting. Knaster asked whether these composants are all homeomorphic, with the obvious exception of the zero composant. Partial results were obtained by Bellamy (1979), Dębski and Tymchatyn (1987), and Aarts and Fokkink (1991). We answer Knaster's question in the affirmative. The main tool is a very simple type of symbolic dynamics for the horseshoe and related continua.
2
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Topological spaces admitting a unique fractal structure

63%
Bandt C., Retta T.
Fundamenta Mathematicae
|
1992
|
tom 141
|
nr 3
257-268
EN
Each homeomorphism from the n-dimensional Sierpiński gasket into itself is a similarity map with respect to the usual metrization. Moreover, the topology of this space determines a kind of Haar measure and a canonical metric. We study spaces with similar properties. It turns out that in many cases, "fractal structure" is not a metric but a topological phenomenon.
3
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Self-affine fractals of finite type

63%
Bandt C., Mesing M.
Banach Center Publications
|
2009
|
tom 84
|
nr 1
131-148
EN
In the class of self-affine sets on ℝⁿ we study a subclass for which the geometry is rather tractable. A type is a standardized position of two intersecting pieces. For a self-affine tiling, this can be identified with an edge or vertex type. We assume that the number of types is finite. We study the topology of such fractals and their boundary sets, and we show how new finite type fractals can be constructed. For finite type self-affine tiles in the plane we give an algorithm which decides whether the tile is homeomorphic to a disk.
4
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On the permeability of submeasures on finite algebras

44%
Bandt C.
Colloquium Mathematicum
|
1978-1979
|
tom 40
|
nr 2
311-318
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