Journal of the Mathematical Society of Japan

Eventual colorings of homeomorphisms

Yuki IKEGAMI, Hisao KATO, and Akihide UEDA

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In this paper, we study some dynamical properties of fixed-point free homeomorphisms of separable metric spaces. For each natural number $p$, we define eventual colorings within $p$ of homeomorphisms which are generalized notions of colorings of fixed-point free homeomorphisms, and we investigate the eventual coloring number $C(f,p)$ of a fixed-point free homeomorphism $f: X \to X$ with zero-dimensional set of periodic points. In particular, we show that if $\dim X$ < $\infty$, then there is a natural number $p$, which depends on $\dim X$, and $X$ can be divided into two closed regions $C_{1}$ and $C_{2}$ such that for each point $x\in X$, the orbit $\{f^{k}(x)\}_{k=0}^{\infty}$ of $x$ goes back and forth between $C_1-C_2$ and $C_2-C_1$ within the time $p$.

Article information

J. Math. Soc. Japan, Volume 65, Number 2 (2013), 375-387.

First available in Project Euclid: 25 April 2013

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Zentralblatt MATH identifier

Primary: 54F45: Dimension theory [See also 55M10] 54H20: Topological dynamics [See also 28Dxx, 37Bxx]
Secondary: 55M10: Dimension theory [See also 54F45] 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space 54C05: Continuous maps

topological dynamics fixed-point free homeomorphism coloring eventual coloring dimension periodic point


IKEGAMI, Yuki; KATO, Hisao; UEDA, Akihide. Eventual colorings of homeomorphisms. J. Math. Soc. Japan 65 (2013), no. 2, 375--387. doi:10.2969/jmsj/06520375.

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