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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
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