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Self-affine curves and sequential machines. (English) Zbl 0937.68072

The paper begins with a brief introduction to a family of \( (m,\alpha)\)-self-affine functions defined by T. Kamae [Japan J. Appl. Math. 3, 271-280 (1986; Zbl 0646.28005)], and to sequential machines and sequential functions according to 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 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.

MSC:

68Q45 Formal languages and automata
54H20 Topological dynamics (MSC2010)
68Q42 Grammars and rewriting systems
26A30 Singular functions, Cantor functions, functions with other special properties
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
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