数理解析研究所講究録
1531 巻 2007 年 140-144140

稠密な周期点集合を持つグラフ写像について
島根大学総合理工学部 横井勝弥 (KATSUYA YOKOI)

1. INTRODUCTION
The purpose of this announcement is to establish my recent resultsfor graph self-maps for which the set of periodic points is dense. M.
Barge and J. Martin [2] showed a structure theorem for maps on theinterval with dense periodic points; that is, the twice iterate of sucha map is topologically mixing on some countable subintervals and isidentical on the other. A similar theorem was proved for tree maps in[7].
We extend the above to graph self-maps (see \S 3). A motivation forstudying graph maps is that higher-dimensional dynamics can often
be reduced to one-dimensional dynamics: this is the case in the studyof the structure of attractors of a diffeomorphism, the quotient maps
generated by maps on manifolds with an invariant foliation of codimension one and the dynamics of pseudo-Anosov homeomorphisms ona surface.
Throughout this paper, by a graph, we mean a connected compact
one-dimensional polyhedron, and a tree is a graph which contains noloops. For a graph , we denote the sets of endpoints and of branch, respectively. A map is a continuous
points of by
andfunction;is the identity map, and for every $n\geq 0,$ $f^{n+1}=f^{n}of$ .
We denote by Fix $(f)$ and Per $(f)$ the sets of fixed points and of periodic
points of , respectively. A subset $K$ of $X$ is invariant under : $Xarrow X$
if $f(K)\subseteq K$ , Int $K$ and Cl $K$ denote the interior and closure of $K$ in$X$ , and the orbit of $x\in X$ underis Orb $f(x)=\{f^{n}(x)|n\geq 0\}$ .
$S,$
denotes the least common multiple ofFor a natural number
the positive integers less than or equal to .
$G$

$G$

$\mathrm{E}(G)$$f$$\mathrm{B}(G)$$f^{0}$

$f$

$f$

$f$

$N_{S}$$S$.
2000 Mathematics Subject Classification.
Key words and phrases. transitive, totally transitive, topologically mixing,$37\mathrm{E}25,37\mathrm{B}20$graph.
The author was partially supported by the Grant-in-Aid for Scientific Research(C) (No. 16540067), the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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2. PRELIMINARIES

An onto map : $X$ $X$ is called (topologically) transitive if any of
the following equivalent conditions holds.
(i) There exists a point with dense orbit.
(ii) Whenever $U,$ are non-empty open sets, there exists an $n\geq 1$such that.
(iii) The only closed invariant set $K$ with Int
is $K=X$ .
We note that if
is transitive for some , then so is .
A map is totally transitive if
is transitive for all $n\geq 1$ . Atransitive map is not always totally transitive. On the other hands, itis well known that for a transitive graph map with periodic points, theset of periodic points is dense. Therefore for such a map , thepowerhas dense periodic points for $n\geq 1$ .
A map : $Xarrow X$ is called topologically mixing if for every pairof non-empty open sets
and , there exists an $N\geq 1$ such thatfor $n\geq N$ . A topological mixing map on a compactumis in general totally transitive. It is also known that a totally transitivegraph map with periodic points is topologically mixing.
R. Roe [7] showed a decomposition theorem for tree maps for which
the set of periodic points is dense. It is slightly -worded here. The
case of interval maps was proved earlier by Marge-Martin [2].
$f$

—

$V$

$f^{-n}(U)\cap V\neq\emptyset$$K\neq\emptyset$$f^{n}$$f$$n$$f$$f^{n}$$f$$n\mathrm{t}\mathrm{h}$$f^{n}$$f$$U$$V$$f^{-n}(U)\cap V\neq\emptyset$$\mathrm{r}$Theorem 2.1 ([7, Theorem 5]). Let : $Tarrow T$ be a tree map for whichthe set of periodic points is dense. Let.
Then there exists a collection (perhaps finite or empty)
ofsubtrees of with disjoint interiors such that(i)for $i\geq 1$ ,(ii):is totally transitive for $i\geq 1$ , and$f$$N_{\mathrm{E}(T)}=\mathrm{L}\mathrm{C}\mathrm{M}\{2,3, \ldots, \neq \mathrm{E}(T)\}$$\{J_{1}, J_{2}, \cdots\}$$T$$f^{N_{\mathrm{E}(T)}}(J_{i})=J_{i}$$f^{N_{\mathrm{E}(T)}}|_{J:}$(iii)$J_{i}arrow J_{i}$$f^{N_{\mathrm{E}(T)}}(x)=x$for$x \in T\backslash \bigcup_{i}J_{i}$.

Remark. His proof of Theorem 5 and Lemma 1 in [7] show totallytransitivity of:above.
$f^{N_{\mathrm{E}(T)}}|_{J_{i}}$$J_{i}arrow J_{i}$Let be a graph, $x\in G$ , and an open connected neighborhood ofin whose closure is a tree. The number of components ofis called the valence of and is denoted by
and we set$\max\{\mathrm{v}(x)|x\in G\}$ . A point of valence
is a branch point and ofvalence 1 is an endpoint.
The .llowing theorem is a direct generalization of [1, Lemma 2] or[6, Corollary 3.2].
$G$

$x$

$U$

$G$

$U\backslash \{x\}$$x$$\mathrm{v}(x)$$\geq 3$$\mathrm{f}\mathrm{o}$$\mathrm{v}(G)=$142
Theorem 2.2 (cf. [6, Corollary 3.2]). Let : $Garrow G$ be a graphand Cl Orb $f(x)=G$ for some $x\in X.$ Let
map satisfying Fix. Then one of the following occurs:in which case Cl Orb $f^{s}(f^{k}(x))=G$ for(i) Cl Orb$s\geq 1$ and $k\geq 0_{r}i.e.,$is totally transitive.
in which case there exists a number(ii) Cl Orb
such that,Cl Orb(a)
is a subgraph of for $0\leq i\leq p-1_{f}$(b) Cl Orb
Int Cl Orb(c) Int Cl Orb
for $0\leq i<$$j\leq p-1$ ,(mod ), and(d) (Cl Orb(e) Cl Orb
for $k\geq 1$ and $0\leq i\leq$
is totally transitive for $0\leq i\leq$$p-1,$ $i.e.,$$f$$(f)\neq\emptyset$$N_{\mathrm{v}(G)}=\mathrm{L}\mathrm{C}\mathrm{M}\{2,3, \ldots , \mathrm{v}(G)\}$$f^{N_{\mathrm{v}(G)}}(x)=G_{f}$$f$$p_{f}$$f^{N_{\mathrm{v}(G)}}(x)\neq G_{f}$$2\leq p\leq \mathrm{v}(G)$$f^{p(f^{i}(x))}$$G= \bigcup_{i=0}^{p-1}$$G$$f^{p(f^{i}(x))}$$f^{p(f^{j}(x))}=\emptyset$$f^{\mathrm{p}(f^{1}(x))}\cap$$f$$f^{\mathrm{p}(f^{i}(x)))}=\mathrm{C}\mathrm{l}\mathrm{O}\mathrm{r}\mathrm{b}_{f^{p}}(f^{i+1}(x))$$p$$f^{\mathrm{p}k(f^{i}(x))}=\mathrm{C}\mathrm{l}\mathrm{O}\mathrm{r}\mathrm{b}_{f^{\mathrm{p}}}(f^{i}(x))$$f|_{\mathrm{C}\mathrm{l}\mathrm{O}\mathrm{r}\mathrm{b}_{f^{\mathrm{p}}}(f^{:}(x))}$$p-1$ .

3. RESULTS
Here is our main theorem.
Theorem 3.1. Let : $Garrow G$ be a graph map for which the set of
periodic points is dense. Then there exist a natural number $N$ anda collection (perhaps finite or empty)of subgraphs ofwith dtsjoint interiors such that(i) $f^{N}(G_{i})=G_{i}$ for $i\geq 1$ ,is totally transitive ( $i.e.$ , topologically mixing)(ii):
for $i\geq 1$ , and
$f$

$G$

$\{G_{1}, G_{2}, \cdots\}$$f^{N}|_{G_{i}}$$G_{i}arrow G_{i}$(iii) $f^{N}(x)=x$ for$x \in G\backslash \bigcup_{i}G_{i}$.

Remark. For simplicity, we used the Roe decomposition theorem fortree maps (Theorem 2.1) in our proof. We are able to prove Theorem 3.1 by use of the Barge-Martin decomposition theorem for intervalmaps [2], [4].

4. EXAMPLES
Example1. Let : $[0,1]arrow[0,1]$ be the map whose graph appears
below, where copies of the small square converge to $\{(0,1)\}$ or $\{(1,0)\}$ ,$f(\mathrm{O})=1$ , and $f(1)=0$ .
which are theThen the closed intervals
projective images of those squares to the first coordinate have that$x\in[0,1]\backslash$$f^{2}(I_{i})=’ J_{i},$: is totally transitive, and $f^{2}(x)=x$ for$f$$J_{;}$$f^{2}|_{J}$$\bigcup_{i}J_{i}$.

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Example 2. Let
be the unit circle on the complex plane. Using
the map : $[0,1]arrow[0,1]$ in Example 1, we define the continuous, where $0\leq\theta\leq 1$ . Put
map : $S^{1}arrow S^{1}$ by
is totally. Then we have that $g^{2}(H_{i})=H_{i},$transitive, and $g^{2}(x)=x$ for.
$S^{1}$$f$$g(e^{2\pi i\theta})=e^{2\pi if(\theta)}$$g$$H_{i}=\{e^{2\pi i\theta}|\theta\in J_{i}\}$$g^{2}|_{H_{i}}$$x \in S^{1}\backslash \bigcup_{i}H_{i}$Example 3. Let
be the bouquet defined by the one-point unionof the unit circle .
andon the origins of the thr\‘ee copies
by the productive coordinate
Here we may write any element of$0\leq\theta\leq 1,$ $j=0,1,2$ .
in Example 2,
Using :, wheredefine : $B_{3}arrow B_{3}$ by
as in Example 2, $i\geq 1$ , and$j=0,1,2$ . Put $K_{i}^{j}=H_{i}\cross\{j\}$ , whereis totally transitive,$j=0,1,2$ . Then we see that
$B_{3}$$S_{0},$$S^{1}$$S_{2}$$S_{1}$$B_{3}$$(e^{2\pi i\theta},j),$$S^{1}arrow S^{1}$

$g$

$h$

$h((e^{2\pi i\theta},j))=(g(e^{2\pi i\theta}),j+1(\mathrm{m}\mathrm{o}\mathrm{d} 3))$

$H_{i}$$h^{6}(K_{i}^{j})=K_{i}^{j},$and $h^{6}(x)=x$ for$x \in B_{3}\backslash \bigcup_{j=0,1,2}\bigcup_{i}K_{i}^{j}$$h^{6}|_{K^{j}}\dot{.}$.

Decomposition theorem does not always hold for general spaces.

Example 4. Let
be the bouquet with -petals generated by the$n\geq
1$like in Example 3. Attach,. Define :unit circle forof the half real line
to the point
for each $n\geq 1$ , the origin of, and the one-dimensional locally finite (non-compact) polyhedron
for $n\geq 1$
is denoted by $B$ . The map : $Barrow B$ is defined by. Then the map has no decomposition
and $\hat{h}(x)=x$ forin the conclusion of our theorem.
$n$$B_{n}$$h_{n}$$B_{n}arrow B_{n}$$h$$\{n\}$$B_{n}$$\mathbb{R}^{\geq 0}$$\hat{h}|_{B_{n}}=h_{n}$$\hat{h}$$x \in B\backslash \bigcup_{n\geq 1}B_{n}$Example 5. Let: $B_{n}arrow B_{n},$ $n\geq 1$ , be as in Example 3. Attach,to the point $\{1/n\}$ of the unit interval
for each $jn\geq 1$ , the origin of$[0,1]$ on condition that the diameter ofis less than or equal to $1/n$ ,and the one-dimensional Peano continuum is denoted by $C$ . The map$h_{n}$$B_{n}$$B_{n}$144
for $n\geq 1$ and $\check{h}(x)=x$ for: $Carrow C$ is defined by. Then the map also has no decomposition in thetheorem.
conclusion$\check{h}|_{B_{n}}=h_{n}$$\check{h}$$x \in C\backslash \bigcup_{n>1}B_{n}$$0\overline{\mathrm{f}}\mathrm{o}\mathrm{u}\mathrm{r}$REFERENCES[1] M. Barge and J. Martin, Chaos, periodicity, and snakelike continua, RtS.
Amer. Math. Soc., 289(1), (1985), 355-365.
[2] M. Barge and J. Martin, Dense periodicity on the interval, Proc. Amer. Math.
Soc., 94 (4), (1985), 731-735.
[3] E. M. Coven and I. Mulvey, Ransitivity and the centre for maps of the circle,
Ergodic Theory Dynam. Systems, 6(1), (1986), 1-8.
[4] E. M. Coven and I. Mulvey, The Barge-Martin decomposition theorem forpointwise nonwandering maps of the interval, Lecture Notes in Math.,
Springer, Berlin 1342, (1988), 100-104.
[5] K. Kuratowski, Topology Vol. I., Academic Press, PWN, Warsaw.
[6] R. P. Roe, Dynamics and in decomposable inverse limit spaces of maps on finitegraphs, Topology Appl., 50(2), (1993), 117-128.
[7] R. P. Roe, Dense periodicity on finite trees, Topology Proc., 19, (1994), 237-248.

DEPARTMENT OF MATHEMATICS, SHIMANE UNIVERSITY, MATSUE, 690-8504,
JAPAN-mail address: yokoiQriko. shimane-u. . jp
$E$$\mathrm{a}\mathrm{c}$