
SIGMA 11 (2015), 084, 21 pages arXiv:1308.3819
https://doi.org/10.3842/SIGMA.2015.084
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems
Michael F. Barnsley ^{a} and Andrew Vince ^{b}
^{a)} Mathematical Sciences Institute, Australian National University, Australia
^{b)} Department of Mathematics, Univesity of Florida, USA
Received June 23, 2015, in final form October 13, 2015; Published online October 16, 2015
Abstract
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 noninteger dimension and comprise a class of deterministic fractal sets. The relationship between the basin and the fast basin of a pointfibred 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 pointfibred 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.
Key words:
iterated function system; fast basins; fractal continuation; fractal manifold.
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