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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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