Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems

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 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.

References in corpus (4)

• The Conley Attractor of an Iterated Function System
• The chaos game on a general iterated function system from a topological point of view
• Groupoid Actions on Fractafolds
• Fractal Tiling

Cited by in corpus (5)

• Tiling iterated function systems and Anderson-Putnam theory
• Conjugacies provided by fractal transformations I : Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces
• SRB measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus
• Central Open Sets Tilings
• Tiling Iterated Function Systems