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On the self-similar Jordan arcs admitting structure parametrization. (Russian, English) Zbl 1117.54052
Sib. Mat. Zh. 46, No. 4, 733-748 (2005); translation in Sib. Math. J. 46, No. 4, 581-592 (2005).
Summary: We study the attractors \(\gamma\) of a finite system of contraction similarities \(S_j\) (\(j = 1,\dots, m\)) in \(\mathbb R^d\) which are Jordan arcs. We prove that if a system \(S\) possesses a structure parametrization \((T, \varphi)\) and \(F(T)\) is the associated family of \(T\) then we have one of the following cases for the identity mapping \(\text{Id}\):
1. \(\text{Id}\) does not belong to the closure of \(F(T)\). Then \(S\) (if properly rearranged) is a Jordan zipper.
2. \(\text{Id}\) is a limit point of \(F(T)\). Then the arc \(\gamma\) is a straight line segment.
3. \(\text{Id}\) is an isolated point of \(\overline{F(T)}\).
We construct an example of a self-similar Jordan curve which implements the third case.
MSC:
54H99 Connections of general topology with other structures, applications
28A80 Fractals
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