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Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2015, том 11
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Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems
Інший ID:
2010 Mathematics Subject Classification: 05B45; 37B50; 52B50
DOI:10.3842/SIGMA.2015.084
DOI:10.3842/SIGMA.2015.084
Посилання:
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.
Підтримка:
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).
Дата:
2015
Переглядів:
576
Завантажень:
296
Анотація:
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.
