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January, 2017 Surface diffeomorphisms with connected but not path-connected minimal sets containing arcs
Hiromichi NAKAYAMA
J. Math. Soc. Japan 69(1): 227-239 (January, 2017). DOI: 10.2969/jmsj/06910227

Abstract

In 1955, Gottschalk and Hedlund introduced in their book that Jones constructed a minimal homeomorphism whose minimal set is connectd but not path-connected and contains infinitely many arcs. However the homeomorphism is defined only on this set. In 1991, Walker first constructed a homeomorphism of $S^1\times \mathbf{R}$ with such a minimal set. In this paper, we will show that Walker's homeomorphism cannot be a diffeomorphism (Theorem 2). Furthermore, we will construct a $C^\infty$ diffeomorphism of $S^1\times \mathbf{R}$ with a compact connected but not path-connected minimal set containing arcs (Theorem 1) by using the approximation by conjugation method.

Citation

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Hiromichi NAKAYAMA. "Surface diffeomorphisms with connected but not path-connected minimal sets containing arcs." J. Math. Soc. Japan 69 (1) 227 - 239, January, 2017. https://doi.org/10.2969/jmsj/06910227

Information

Published: January, 2017
First available in Project Euclid: 18 January 2017

zbMATH: 1393.37056
MathSciNet: MR3597553
Digital Object Identifier: 10.2969/jmsj/06910227

Subjects:
Primary: 37E30
Secondary: ‎37B05‎, 37B45

Rights: Copyright © 2017 Mathematical Society of Japan

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Vol.69 • No. 1 • January, 2017
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