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Self-similar centralizers of circle maps. (English) Zbl 0991.37020
The author studies the self-similarity of certain expanding maps $$f:S^1\to S^1$$ of the unit circle. $$f\in \text{Imm}^n (S^1)$$ if $$f'(z)\neq 0$$ for all $$z\in S^1$$ (endowed with the $$C^n$$-topology). Let $$Z(f)$$ denote the centralizer of $$f$$, then $$Z(f)$$ is self-similar if $$g\in Z(f)$$ implies that $$Z(f)$$ and $$Z(g)$$ are isomorphic. It is shown that if $$f$$ is an expanding, orientation preserving map in $$\text{Imm}^n (S^1)$$ of degree $$p$$, and if there exists $$g\in Z(f)$$ of degree $$q$$ ($$p$$ and $$q$$ coprime), then $$Z(f)$$ is not self-similar.
##### MSC:
 37E10 Dynamical systems involving maps of the circle 28A80 Fractals
##### Keywords:
circle maps; self-similarity; expanding maps; centralizer
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