Let X denote a compact metric space with metric d, and let f : X → X denote a continuous self-map on X . For any subset E of X , we let Cl(E) denote the closure of E. Following [1], we denote by { ,X} an iterated function system, or IFS, onX . That is, is a finite family { f1, . . . , fm} of continuous self-maps on X . In this paper we do not consider the case in which is an infinite family. Le...

Valuations — morphisms from (Σ ; ;e) to ((0;∞); ;1) — are a generalization of Bernoulli morphisms introduced in [10]. Here, we show how to generalize the notion of entropy (of a language) in order to obtain new formulae to determine the Hausdorff dimension of fractal sets (also in Euclidean spaces) especially defined via regular (ω-)languages. By doing this, we can sharpen and generalize earlie...

We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domain-theoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic I...

A fast method to generate fractal imagery is presented. Iterated Function Systems (IFS) are based on repeatedly copying transformed images. We show that this can be directly translated into standard graphics operations: Each image is generated by texture mapping and blending copies of the previous image. Animations of dynamic IFS codes are generated at fifty frames per second on a notebook PC, ...

This thesis presents an approach which links logic programs and iterated function systems. The work was motivated by the discovery of self-similar structures during investigations of the TP-operator. This operator, associated with any program P , was previously embedded into the real numbers. Therefore, it could be plotted as a graph in R. It is shown that for a large class of logic programs th...

In this small note an iterated function system on the space of distribution functions is built. The inverse problem is introduced and studied by convex optimization problems. Applications of this method to approximation of distribution functions and estimation are presented. Résumé. Dans cette petite note un système de fonction itéré sur l’espace de fonctions de repartition est construit. Le pr...

This paper presents a genetic algorithm used to infer Iterated Function Systems (IFSs) for approximation of 1D and 2D data sets. First, we give an introduction to IFSs. Secondly, we discuss the encoding of 1D and 2D IFSs in the form of chromosomes, and present genetic operators to work upon these representations. The performance of the genetic algorithm is evaluated on four 1D and two 2D test s...

The sequence {Fn} is called the iterated function system corresponding to {fn}. For readability, we denote by F(Ω, X) the set of all iterated function systems made from functions in Hol(Ω, X). By Montel’s theorem (see for example [2]), the sequence Fn is a normal family, and every convergent subsequence converges uniformly on compact subsets of Ω to a holomorphic function F ∈ Hol(Ω, X). The acc...

Suppose you had no ideas what fractals were, and you were asked to model objects like clouds, grass, plants, etc.. What else could you use to define the geometries of these objects? Although polynomials can easily define objects with smooth geometry, they are pretty useless if you want to model complex objects like grass or plants, since those possess infinitely non smooth, highly structured ge...