数理解析研究所講究録
第 1781 巻 2012 年 97-102

97

Colorings and eventual colorings
池上 勇気 (Yuki Ikegami) , 加藤 久男 (Hisao Kato) , 上田 晃秀 (Akihide Ueda)

筑波大学大学院数理物質科学研究科

Institute of Mathematics, University of Tsukuba.
Abstract. 本稿では、 可分距離空間上における不動点を持たない同相写像の力学的な性質の幾つかを扱う。与えられた周期点の集合が零次元である不動点を持たない同相写像 : $Xarrow X$ と、 任意の自然数 に対して、 coloring number を一般化した eventual
coloring number $C(f,p)$ を定義しその性質を調べる。特に、 空間 $X$ が有限次元である場
$f$

$p$

合、 $X$ が 2 つの閉集合ととに分割でき、 $X$ のすべての元が $f$ を 回施す間にの間を行き来するような $X$ の次元に依存した自然数 が存在することを紹介する。
$C_{1}$$C_{2}$$C_{1}$$p$$C_{2}$$p$1Introduction

All spaces are assumed nonempty separable metric spaces and all maps are continuousfunctions. Let be the set of all natural numbers, i.e., $\mathbb{N}=\{1,2,3, \cdots\}$ . For a (separable$N$metric) space $X,$ $\dim X$ denotes the topological dimension of $X$ . Let $P(f)$ be the set of
all periodic points of map $f$ : $Xarrow X$ , i.e.,$P(f)=\{x\in X|f^{j}(x)=x$ for some$j\in \mathbb{N}\}$.

Let: $Xarrow X$ be a fixed-point free closed map of a separable metric space $X$ .
In this paper, we assume that all maps are closed maps. A subsetof $X$ is called acolor (see [10]) of provided that $f(C)\cap C=\emptyset$ . Note that $f(C)\cap C=\emptyset$ if and only
if $C\cap f^{-1}(C)=\emptyset$ . We say a cover of $X$ is a coloring of if each element $C$ of is acolor of . The minimal cardinality $C(f)$ of closed (or open) colorings of is called thecoloring number of . Many mathematicians have investigated the coloring number $C(f)$(see [1-4], [6] and [8-10]).
$f$

$C$

$f$

$C$

$C$

$f$

$f$

$f$

$f$

Theorem 1.1. (Aarts, Fokkink and Vermeer [1]) Let : $Xarrow X$ be a fixed-point free
involution of a (separable) metric space $X$ with $\dim X=n<\infty$ . Then $C(f)\leq n+2$ .
$f$Theorem 1.2. (Aarts, Fokkink and Vermeer [1]) Let : $Xarrow X$ be a fixed-point freehomeomorphism of a (separable) metric space $X$ with $\dim X=n<\infty$ . Then $C(f)\leq$$n+3$ .
$f$

Now, we will define more general notion of color. Let : $Xarrow X$ be a fixed-point
free map of a space $X$ and. A subset $C$ of $X$ is eventually colored within ofif
. It is easy to see that $C$ is a color of if and only if $C$ is eventuallycolored within 1. Then we have the following proposition.
$f$$p\in \mathbb{N}$$\bigcap_{i=0}^{p}f^{-i}(C)=\emptyset$

$p$

$f$

$f$

98Proposition 1.3. Let : $Xarrow X$ be a fixed-point free map of a sepamble metric space$X$ and. Then the followings hold.
(1) A subset of $X$ is eventually colored within of if and only if each point $x\in C$
wanders off within i. e., for each $x\in C_{f}f^{i}(x)\not\in C$ with some $i\leq p$ .
then is eventually colored(2) If a subset of $X$ satisfies the.
within of(3) If is an injective map, then a subsetof $X$ is eventually colored within of if.
and only if satisfies the condition$f$$p\in \mathbb{N}$

$C$

$p$

$C$

$C$

$p$

$f$

$p_{f}$$C$$\omega ndition\bigcap_{i=0}^{p}f^{i}(C)=\emptyset_{f}$

$f$

$C$

$f$

$C$

$f$

$p$

$\bigcap_{i=0}^{p}f^{i}(C)=\emptyset$

Remark. In general, the converse assertion of (2) in the proposition above is not true.
Let $X=\{a, b, c\}$ be a set consisting three points and let : $Xarrow X$ be the map definedby $f(a)=b,$ $f(b)=c,$ $f(c)=b$ . Then $C=\{a, b\}$ is eventually colored within 2 of , but
$f$

$f$

$\bigcap_{i=0}^{p}f^{i}(C)\neq\emptyset(p\in N)$.

Next, we define the eventual coloring number $C(f,p)$ as follows. A cover of $X$ iscalled an eventual coloring within if each element $C$ of is eventually colored within. The minimal cardinality $C(f,p)$ of all closed (or open) eventual colorings within iscalled the eventual coloring number of within . Note that $C(f, 1)=C(f)$ . If there
is some $p\in N$ with $C(f,p)<\infty$ , we say that is eventually colored. Similarly, we canconsider the index $C^{+}(f,p)$ defined by
$C$

$C$

$p$

$p$

$p$

$f$

$p$

$f$

$\min${ $|C|;C$ is a closed (open) cover of$X$such that for each$C \in C,\bigcap_{i=0}^{p}f^{i}(C)=\emptyset$}.

By the definitions, we see that $C(f,p)\leq C^{+}(f,p)$ . In section 3, we show that $C(f,p)=$$C^{+}(f,p)$ if $X$ is compact.
In this paper, we need the following notions. A finite cover of $X$ is a closed partition of$X$ provided that each element $C$ ofand $C\cap C’=bd(C)\cap bd(C’)$ foris closed, int$C^{l}\in
C$$X$$C,$any. Let be a collection of subsets of a space with $\dim X=n<\infty$ . Thenwith $|S|\leq n+1$ ,we say that is in general position in $X$ (see [7]) provided that if$\dim(\cap\{S|S\inS\})\leq
n-|S|$. By a swelling of a familyof subsets of a space
then$X$ , we mean any familyof subsets of $X$ such that $A_{s}\subset B_{s}(s\in S)$ and for every$s_{m}\in S$ ,finite set of indices$C$$C$$(C)\neq\emptyset$$\mathcal{B}$$S\subset \mathcal{B}$$\mathcal{B}$$\{A_{S}\}_{s\in S}$$\{B_{S}\}_{\epsilon\in S}$$s_{1},$ $s_{2},$$\ldots,$$\bigcap_{i=1}^{m}A_{s}:\neq\emptyset$

if and only if$\bigcap_{i=1}^{m}B_{s_{i}}\neq\emptyset$.

ifof $X$ is a shrinking ofConversely, for any coverof $X$ , a cover$A_{s}\subset B_{s}(s\in S)$ . The following facts are well-known;of closed subsets of a space $X,$
(1) for any locally finite collection
has a swellingconsisting of open subsets of $X$ (e.g., see [10, Proposition 3.2.1]) and(2) for any open coverof $X,$
has a closed shrinking cover of $X$ (e.g., see [10,Proposition A.7.1] .
Hence we see that if : $Xarrow X$ is a closed map and a closed finite cover of $X$ is an
eventual coloring of , then we can find an open swelling of which is an eventual
coloring of .
$\{A_{s}\}_{s\in S}$$\{B_{s}\}_{s\in S}$$\{B_{s}\}_{\epsilon\in S}$$\mathcal{F}$$\mathcal{F}$$\mathcal{U}$$\mathcal{U}$$)$

$f$

$f$

$f$

$\mathcal{B}$$C$$\mathcal{B}$99

Two indices which evaluate eventual coloring numbers

2

In this section, we will define indices. First, for each $n=0,1,2,$ ,andand each $k=0,1,2,$ $n+1$ , we define the index
as follows: Put $\varphi_{n}(0)=1$ . For
each $k=1,2,$ $n+1$ , by induction on we define the index
by$\varphi_{n}(k)$$\tau_{n}(k)$$\ldots$$\varphi_{n}(k)$$\ldots,$$k$$\varphi_{n}(k)$$\ldots,$$\varphi_{n}(k)=2\varphi_{n}(k-1)+[n/(n+2-k)]\cdot(\varphi_{n}(k-1)+1)$,

where $[x]= \max\{m\in NU\{0\}|m\leq x\}$ for $x\in[0, \infty)$ . Next, we will consider anotherindex
defined by$\tau_{n}(k)$$\tau_{n}(k)=k(2n+1)+1$

for each $n=0,1,2,$ , and each $k=0,1,2,$
theorem in this paper.
$\ldots$$\ldots,$Thus, we obtain following main$n+1$ .

Theorem 2.1. ([5, Theorem 2.3, Theorem 2.6]) Let : $Xarrow X$ be a fixed-point freehomeomorphism of a sepamble metric space $X$ with $\dim X=n<\infty$ . If $\dim P(f)\leq 0$ ,then$f$$C(f, \min\{\varphi_{n}(k), \tau_{n}(k)\})\leq n+3-k$

for each $k=0,1,2,$$\ldots,$$n+1$ .

Remark. If we do not assume $\dim P(f)\leq 0$ , the above theorem is not true. Let:be the antipodal map of the n-dimensional sphere . Note that $P(f)=S^{n}$and $C(f,p)=C(f, 1)=n+2$ for any.
In fact we have the following tables of values of two indices.
$f$$S^{n}arrow S^{n}$$S^{n}$$p\in \mathbb{N}$$\varphi_{n}(k)$$\tau_{n}(k)$100. This way is repainting
is similar to way of
The way of the construction ofa color one by one. Thus, we expect that repainting many colors all at once will bring ato us. But this way is very complicated.
better index thanand$\varphi_{n}(k)$$\tau_{n}(k)$$\varphi_{n}(k)$$\tau_{n}(k)$Now we have the following general problem for eventual coloring numbers.
Problem 2.2. For each $n\geq 0$ and each $1\leq k\leq n+1$ , determine the minimal numberof natural numbers satisfying the condition; if : $Xarrow X$ is any fixed-point freehomeomorphism of a sepamble metric space $X$ such that $\dim X=n$ and $\dim P(f)\leq 0_{f}$then $C(f,p)\leq n+3-k$ .

$\mu_{n}(k)$

$f$

$p$

By comparing two indicesproblem.

$\varphi_{n}(k)$and$\tau_{n}(k)$, we have a partial answer to the above

Corollary 2.3. Suppose that
: $Xarrow X$ is a fixed-point free homeomorphism of asepamble metric space $X$ and $\dim P(f)\leq 0$ .
(1) If $\dim X=0$ , then $C(f, 2)=2$ .
(2) If $\dim X=1_{f}$ then $C(f, 7)=2$ .
(3) If $\dim X=2$ , then $C(f, 16)=2$ .
(4) If $\dim X=3$ , then $C(f, 29)=2$ .
(5) If $\dim X=4_{f}$ then $C(f, 46)=2$ .
In other words, $\mu_{0}(1)=2,$ $\mu_{1}(2)\leq 7,$ $\mu_{2}(3)\leq 16,$ $\mu_{3}(4)\leq 29$ and $\mu_{4}(5)\leq 46$ .
$f$

In addition, we have the following result which is the case $C(f,p)=2$ .
Corollary 2.4. Let: $Xarrow X$ be a fixed-point free homeomorphism of a sepamble$X$$\dim
X=n<\infty$ . If $\dim P(f)\leq 0$ , then there is some $p\in N$ withmetric spacewith$p \leq\min\{\varphi_{n}(n+1), \tau_{n}(n+1)\}$ such that$f$$C(f,p)=2$ .
$C_{2}(i.e., X=C_{1}\cup C_{2})$ andIn other words, $X$ can be divided into two closed subsets
there is somesuch that if $x\in C_{i}(i\in\{1,2\})$ , there is a strictly increasing sequenceof natuml numbers such that $1\leq n_{x}(1)\leq p,$ $n_{x}(k+1)-n_{x}(k)\leq p$ and if$j\in\{1,2\}$ with $j\neq i$ , then$C_{1},$$p\in \mathbb{N}$$\{n_{x}(k)\}_{k=1}^{\infty}$$f^{n_{x}(k)}(x)\in C_{j}-C_{i}(k:odd),$3$f^{n_{x}(k)}(x)\in C_{i}-C_{j}(k.\cdot even)$.

Eventual coloring numbers on compact metric spaces

In this section, we consider eventual coloring numbers of fixed-point free maps ofcompact metric spaces. Let $X$ be a compact metric space and let : $Xarrow X$ be a map.
Consider the inverse limit $(X, f)$ of , i.e.
$f$

$f$

$(X, f)=\{(x_{i})_{i=0}^{\infty}|x_{i}\in X,$ $f(x_{i})=x_{i-1}$

for$i \in N\}\subset X^{\infty}=\prod_{j=0}^{\infty}X_{j}$.

101Then we have the shift homeomorphism $\tilde{f}:(X, f)arrow(X, f)$ oftion : $(X, f)arrow X_{j}=X(j\geq 0)$ defined by$f$and the natural projec-$p_{j}$$\tilde{f}((x_{i})_{i=0}^{\infty})=(f(x_{i}))_{i=0}^{\infty},$$p_{j}((x_{i})_{i=0}^{\infty})=x_{j}$.

Note that. We see that if : $Xarrow X$ is a fixed-point free map of a compact
metric space $X$ , then : $(X, f)arrow(X, f)$ is a fixed-point free homeomorphism. By amodification of the proof of [1, Theorem 6], we have the following theorem which is amore precise result than [1, Theorem 6].
$p_{j}\cdot\tilde{f}=f\cdot p_{j_{-}}$

$f$

$f$

. Let : $Xarrow X$ be a fixed-point free map of a compact metric space $X$: $(X, f)arrow(X, f)$ be the shift homeomorphism of . Then for,$Theorem\sim 3.1$

and let

$f$

$f$

$f$

$C(f,p)=C^{+}(f,p)=C(\tilde{f,}p)$$p\in \mathbb{N}$.

Corollary 3.2. (cf. [1, Theorem 6]) Let: $Xarrow X$ be a fixed-point free map of a
$X$compact metric spacewith $\dim X=n<\infty$ . If $\dim P(f)\leq 0$ , then there is $p\in N$ with$p \leq\min\{\varphi_{n}(k), \tau_{n}(k)\}$ such
that
$f$

$C(f,p)\leq n+3-k$

for each $k=0,1,2,$$\ldots,$$n+1$ .

Example. There are a (zero-dimensional) separable metric space $X$ and a fixed-point
free map : $Xarrow X$ such that $\dim P(f)\leq 0$ and
$f$

(1)
(2)
(3)

$f$

$f$
$f$

is closed,
is finite-to-one, andcannot be eventually colored within any $p\in$ N.

Remark. In the statement of Theorem 1.2, “a separable metric space $X$ “ can bereplaced with “a paracompact space $X$ ” (see [M. A. van Hartskamp and J. Vermeer, On
colorings of maps, Topology and its Applications 73 (1996), 181-190] . Hence Theorem
2.1 is also true for the case that $X$ is a paracompact space.
$)$

102

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Yuki Ikegami
Institute of Mathematics
University of Tsukuba
Ibaraki, 305-8571 Japan
Hisao Kato

Institute of Mathematics
University of Tsukuba
Ibaraki, 305-8571 Japane-mail: hkato@math.tsukuba.ac.jp
Akihide Ueda
Institute of Mathematics
University of Tsukuba
Ibaraki, 305-8571 Japan