Self-affine curves and sequential machines
type: Item
instance of: scholarly article
title: Self-affine curves and sequential machines
zbMATH Open document ID: 0937.68072
author: Anna Rodenhausen
author: Heinz-Otto Peitgen
author: G. Skordev
published in: Real Analysis Exchange
publication date: 2000-05-22T00:00:00Z
review text: The paper begins with a brief introduction to a family of \( (m,\alpha)\)-self-affine functions defined by \textit{T. Kamae} [Japan J. Appl. Math. 3, 271-280 (1986; Zbl 0646.28005)], and to sequential machines and sequential functions according to \textit{S. Eilenberg} [Automata, Languages, and Machines, Vol. A, Academic Press, New York (1985; Zbl 0317.94045)], who used them to construct some fractal curves. The authors of this paper generalize the notion of self-affinity in several steps. First they define \((m,L)\)-self-affine curves, \(m \in \mathbb N,\^^ML: {\mathbb R}^2 \to {\mathbb R}^2\) is a linear expanding map; still an \(m\)-adic representation is used. It is shown here that this family of curves coincides with a class of recurrent curves considered by \textit{M. F. Dekking} [Adv. Math. 44, 78-104 (1982; Zbl 0495.51017)]. Next steps are based on the classical Cantor representation of reals in the unit interval corresponding to a substitution of non-constant length. Authors describe curves which are self-affine with respect to a linear expanding map \(L\), later to a finite number of linear expanding maps. Finally a problem of fractal interpolation is considered. Authors show that the graph corresponding to an interpolation function is generated by a sequential machine and they give the suitable definitions of self-affinity corresponding to such functions. For each type of self-affinity authors prove that curves generated by a consistent sequential machine (corresponding to this type) are self-affine, and conversely, each self-affine curve is generated by some sequential machine. Many classical examples are discussed here to illustrate mentioned types of self-affinity.
Mathematics Subject Classification ID: 68Q45
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 68Q42
Mathematics Subject Classification ID: 26A30
Mathematics Subject Classification ID: 26A27
zbMATH DE Number: 1450281
zbMATH Keywords: self-affine functions
zbMATH Keywords: self-affine curves
zbMATH Keywords: sequential machines
zbMATH Keywords: sequential functions
zbMATH Keywords: finite automata
zbMATH Keywords: L-systems
zbMATH Keywords: iterated function systems
reviewed by: Magdalena Hykšová
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1450281