Theory and Applications of Fractal Tops

Bulletin of Mathematical Sciences, 2013

Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor-repeller pairs and the Conley "landscape picture" for an IFS.

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.

Discrete & Computational Geometry, 2014

A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be constructed by this method. These tilings can be used to extend a fractal transformation defined on the attractor of a contractive IFS to a fractal transformation on the entire space upon which the IFS acts.

In this paper we draw attention to some recent advances in fractal geometry and point out several ways in which they apply to digital imaging. Simple applications in- clude a method for animating backgrounds in the produc- tion of synthetic content, including seascapes, forests, and skies; a novel low-cost technique for creating ani- mated talking heads with unique look-and-feel; and the sharing of engaging graphics, at low bandwidth, between wireless devices such as cellphones. These advances make use of an addressing system which may be associated with the "top" of the attractor of an iterated function system (IFS). Previous computer graphics applications of IFS theory have focused on mod- els based on the attractors and the invariant measures of IFSs. The addressing system enables the establish- ment of mappings between attractors; it is these trans- formations, rather than the attractors themselves, that underlie the digital imaging ideas introduced here.

Fractals in Multimedia, 2002

2013

Abstract. We define and exemplify the continuations and the fast basin of an attractor of an IFS. Then we extend the standard symbolic IFS theory, concerning the dynamics of a contractive IFS on its attractor, to a symbolic description of the dynamics of a invertible IFS on a set that contains the fast basin of a point-fibred attractor. We use this description to define the fractal manifold, a new topological invariant, associated with a point-fibred attractor of the IFS. We establish relationships between the fractal manifold, the fast basin, and the set of continuations of an attractor of an IFS. We establish how sections of projections, from code space to an attractor, can be extended to yield sections of projections from code space to the f-manifold and to the basin. We use these sections to construct transformations between fractal manifolds and show how their projections extend fractal transformations between at-tractors to corresponding transformations between fast basins of ...

2021

The main result of this paper states that for a given countable system of data ∆, there exists a countable iterated function system consisting of Rakotch contractions, such that its attractor is the graph of a fractal interpolation function corresponding to ∆. In this way, on the one hand, we generalize a result due to N. Secelean (see The fractal interpolation for countable systems of data, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 14 (2003), 11–19) by considering countable systems consisting of Rakotch contractions rather than Banach contractions. On the other hand, we generalize a result due to S. Ri (see A new idea to construct the fractal interpolation function, Indag. Math., 29 (2018), 962-971) by considering countable (rather than finite) systems consisting of Rakotch contractions. Some exemplifications are provided.

2018

The geometric modeling of fractal objects is a difficult process. An important class of complex objects in nature is trees, plants, clouds, mountains, etc. These objects cannot be satisfactorily described or acceptable in quality or quantity using conventional geometry. Diverse techniques are at present being investigated for modeling these complex objects. The development of a new approach to computing fractals is being taken up by us, known as Iterated Function Systems. Any set of linear maps (affine transformations) and an associated set of probabilities determines an Iterated Function System (IFS). IFS description forms, through a set of simple geometric transformations, a basic set of tools for interactive image construction. This paper presents the role of IFS in geometric modeling of fractal objects.

We investigate chaotic and multi-fractal properties of a two parameter map of the unit interval onto itself-the Kim-Kong map. These results are compared with similar properties in well known one parameter maps of the unit interval onto itself.