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

TL;DR

This paper introduces the concept of fast basins and branched fractal manifolds associated with attractors of iterated function systems, providing new insights into their topology and geometry.

Contribution

It presents the novel concept of branched fractal manifolds to analyze the topology and geometry of fast basins of IFS attractors, inspired by analogies with analytic continuation.

Findings

Fast basins can have non-integer dimensions.

Branched fractal manifolds are constructed from extended code spaces.

These manifolds are unions of nonhomeomorphic generalized fractal blowups.

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.