an:00120181
Zbl 0769.58037
Handel, Michael
There are no minimal homeomorphisms of the multipunctured plane
EN
Ergodic Theory Dyn. Syst. 12, No. 1, 75-83 (1992).
00011016
1992
j
37A99 58D05
periodic points; minimal homeomorphisms; multipunctured plane
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.
Y.Kozai (Tokyo)