## Journal of the Mathematical Society of Japan

### Eventual colorings of homeomorphisms

#### Abstract

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

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

Dates
First available in Project Euclid: 25 April 2013

https://projecteuclid.org/euclid.jmsj/1366896638

Digital Object Identifier
doi:10.2969/jmsj/06520375

Mathematical Reviews number (MathSciNet)
MR3055590

Zentralblatt MATH identifier
1275.54020

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1366896638

#### References

• J. M. Aarts, R. J. Fokkink and H. Vermeer, Variations on a theorem of Lusternik and Schnirelmann, Topology, 35 (1996), 1051–1056.
• J. M. Aarts, R. J. Fokkink and H. Vermeer, Coloring maps of period three, Pacific J. Math., 202 (2002), 257–266.
• A. Błaszczyk and D. Y. Kim, A topological version of a combinatorial theorem of Katětov, Comment. Math. Univ. Carolin., 29 (1988), 657–663.
• E. K. van Douwen, $\beta X$ and fixed-point free maps, Topology Appl., 51 (1993), 191–195.
• A. Krawczyk and J. Steprāns, Continuous colourings of closed graphs, Topology Appl., 51 (1993), 13–26.
• J. Kulesza, Zero-dimensional covers of finite-dimensional dynamical systems, Ergodic Theory Dynam. Systems, 15 (1995), 939–950.
• L. Lusternik and L. Schnirelman, Topological Methods in Variational Calculus (Russian), Moscow, 1930.
• J. van Mill, Easier proofs of coloring theorems, Topology Appl., 97 (1999), 155–163.
• J. van Mill, The Infinite-Dimensional Topology of Function Spaces, North-Holland Math. Library, 64, North-Holland Publishing Co., Amsterdam, 2001.
• H. Steinlein, On the theorems of Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk, Canad. math. Bull., 27 (1984), 192–204.