Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems
The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a class of deterministic fractal sets. The relationship betwe...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2015 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2015
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147148 |
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| Cite this: | Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems / M.F. Barnsley, A. Vince // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 17 назв. — англ. |
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Barnsley, M.F. Vince, A. 2019-02-13T17:40:20Z 2019-02-13T17:40:20Z 2015 Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems / M.F. Barnsley, A. Vince // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 17 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05B45; 37B50; 52B50 DOI:10.3842/SIGMA.2015.084 https://nasplib.isofts.kiev.ua/handle/123456789/147148 The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a class of deterministic fractal sets. The relationship between the basin and the fast basin of a point-fibred attractor is analyzed. To better understand the topology and geometry of fast basins, and because of analogies with analytic continuation, branched fractal manifolds are introduced. A branched fractal manifold is a metric space constructed from the extended code space of a point-fibred attractor, by identifying some addresses. Typically, a branched fractal manifold is a union of a nondenumerable collection of nonhomeomorphic objects, isometric copies of generalized fractal blowups of the attractor. We thank Alan Carey for interesting discussions and helpful comments. We thank Louisa Barnsley for technical help and suggestions. We thank Krystof Le´sniak, who worked with us on the first version of this paper, namely [3], for many comments and suggestions. We thank the anonymous referees for helpful comments. This work was partially supported by a grant from the Simons Foundation (#322515 to Andrew Vince). It was also partially supported by a grant from the Australian Research Council (#DP130101738). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems Article published earlier |
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| title |
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems |
| spellingShingle |
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems Barnsley, M.F. Vince, A. |
| title_short |
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems |
| title_full |
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems |
| title_fullStr |
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems |
| title_full_unstemmed |
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems |
| title_sort |
fast basins and branched fractal manifolds of attractors of iterated function systems |
| author |
Barnsley, M.F. Vince, A. |
| author_facet |
Barnsley, M.F. Vince, A. |
| publishDate |
2015 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a class of deterministic fractal sets. The relationship between the basin and the fast basin of a point-fibred attractor is analyzed. To better understand the topology and geometry of fast basins, and because of analogies with analytic continuation, branched fractal manifolds are introduced. A branched fractal manifold is a metric space constructed from the extended code space of a point-fibred attractor, by identifying some addresses. Typically, a branched fractal manifold is a union of a nondenumerable collection of nonhomeomorphic objects, isometric copies of generalized fractal blowups of the attractor.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147148 |
| citation_txt |
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems / M.F. Barnsley, A. Vince // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 17 назв. — англ. |
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2025-12-01T11:07:49Z |
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2025-12-01T11:07:49Z |
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