Journal ArticleDOI
Iterated function systems and the global construction of fractals
TLDR
In this paper, a general framework for the exactly computable moment theory of p -balanced measures for hyperbolic i.f.ss and of probabilistic mixtures of iterated Riemann surfaces is presented.Abstract:
Iterated function systems (i. f. ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i. f. ss and occur as the supports of probability measures associated with functional equations. The existence of certain ‘ p -balanced’ measures for i. f. ss is established, and these measures are uniquely characterized for hyperbolic i. f. ss. The Hausdorff—Besicovitch dimension for some attractors of hyperbolic i. f. ss is estimated with the aid of p -balanced measures. What appears to be the broadest framework for the exactly computable moment theory of p -balanced measures — that of linear i. f. ss and of probabilistic mixtures of iterated Riemann surfaces — is presented. This extensively generalizes earlier work on orthogonal polynomials on Julia sets. An example is given of fractal reconstruction with the use of linear i. f. ss and moment theory.read more
Citations
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Journal ArticleDOI
Image coding based on a fractal theory of iterated contractive image transformations
TL;DR: The author proposes an independent and novel approach to image coding, based on a fractal theory of iterated transformations, that relies on the assumption that image redundancy can be efficiently exploited through self-transformability on a block-wise basis and approximates an original image by a Fractal image.
Journal ArticleDOI
Nonlinear prediction of chaotic time series
TL;DR: Numerical techniques are presented for constructing nonlinear predictive models directly from time series data and scaling laws are developed which describe the data requirements for reliable predictions.
Journal ArticleDOI
Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing
TL;DR: In this article, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible and immiscible displacement processes in reservoir rocks are reviewed and discussed, and two different modeling approaches to these phenomena are compared.
Journal ArticleDOI
Estimating fractal dimension
TL;DR: The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of Fractals to nonlinear dynamical systems, and to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor.
Book
Fractal Image Compression
TL;DR: Data compression with fractals is an approach to reach high compression ratios for large data streams related to images, at a cost of large amounts of computation.
References
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Book
The Fractal Geometry of Nature
TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
Journal ArticleDOI
An invariant measure for rational maps
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On the uniqueness of the maximizing measure for rational maps
TL;DR: In this article, it was shown that analytic endomorphisms of the Riemann sphere do not have a unique maximizing measure, and the existence of such a measure is proved.
Journal ArticleDOI
Orthogonal polynomials associated with invariant measures on Julia sets
Book
Classical Galois Theory, With Examples
TL;DR: In this paper, the Galois group is used to solve a polynomial solvable equation of degree $n$ in fixed fields, where the solvability of equations has been shown to be solvable by groups of automorphisms.