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There are no minimal homeomorphisms of the multipunctured plane. (English) Zbl 0769.58037
The author proves the following theorem and its corollary.
Theorem. Suppose that \(f: S^ 2 \to S^ 2\) is an orientation-preserving homeomorphism of the two-dimensional sphere and that \(\text{Fix}(f)\) is a finite set containing at least three points. If \(f\) has a dense orbit then the number of periodic points of period \(n\) for some iterate of \(f\) grows exponentially in \(n\).
Corollary. There are no minimal homeomorphisms of the multipunctured plane \(\mathbb{R}^ 2\setminus K\) where \(K\) is a finite set with at least two points.
Reviewer: Y.Kozai (Tokyo)

37A99 Ergodic theory
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
Full Text: DOI
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