Continua as minimal sets of homeomorphisms of $S^2$

Shigenori Matsumoto; Hiromichi Nakayama

doi:10.4171/lem/57-3-5

JournalslemVol. 57, No. 3/4pp. 373–392

Continua as minimal sets of homeomorphisms of $S^{2}$

Shigenori Matsumoto
Nihon University of Science and Technology, Tokyo, Japan
Hiromichi Nakayama
Aoyama Gakuin University, Kanagawa, Japan

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Abstract

Let $f$ be an orientation preserving homeomorphism of $S^{2}$ which has a continuum $X$ as a minimal set. Then there are exactly two connected components of $S^{2} ∖ X$ which are left invariant by $f$ and all the others are wandering. The Carathéodory rotation number of an invariant component is irrational.

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Shigenori Matsumoto, Hiromichi Nakayama, Continua as minimal sets of homeomorphisms of $S^{2}$ . Enseign. Math. 57 (2011), no. 3/4, pp. 373–392

DOI 10.4171/LEM/57-3-5