数理解析研究所講究録
1531 巻 2007 年 134-139134

グラフ写像の強推移性について
島根大学総合理工学部

横井勝弥 (KATSUYA YOKOI)

1. INTRODUCTION
The purpose of this r\’esum6 is to describe (strong) transitivity properties for graph self-maps in my recent works. W. Parry [7] pointed
out a sufficient condition for the existence of a special measure on asymbolic dynamics, which has a close relation to a linearization of thedynamics on intervals. Then, as an application, he introduced the concept of strong transitivity that is one of conditions under which aninterval map is conjugate to a uniformly piecewise linear map [7, \S 5,\S 6]. E. Coven and I. Mulvey [6, Theorem and ] stated the relation
between transitivity and strong transitivity properties for interval (orcircle) self-maps.
We extend the above relation to graph self-maps (see \S 3). A motivation for studying graph maps is that higher-dimensional dynamics canoften be reduced to one-dimensional dynamics.
Throughout this paper, by a graph, we mean a connected compact
one-dimensional polyhedron, and a tree is a graph which contains noloops. We also assume that any graph is endowed with a metric ; we$x\in G,$
to be the set of points of whose distancedefinedenote the sets of branch points
andis less than .
and of endpoints of , respectively. A map is a continuous functionis the identity map, and for every $n\geq 0$ ,from a space $X$ to itself;. We denote by Fix $(f)$ and Per $(f)$ the sets of fixed points
and of periodic points of , respectively. For a subset $K$ of $X$ , Int $K$
and Cl $K$ denote the interior and closure of $K$ in $X$ .
$\mathrm{C}$$\mathrm{B}$

$G$

$d$

$G$

$\epsilon>0$$\mathrm{B}(x;\epsilon),$$\mathrm{b}\mathrm{o}\mathrm{m}x$$\mathrm{B}(G)$$\epsilon$$\mathrm{E}(G)$

$G$

$f$

$f^{0}$$f^{n+1}=f^{n}\mathrm{o}f$$f$

2. STRONG TRANSITIVITY
An onto map : $Xarrow X$ is called (topologically) transitive if any of
the following equivalent conditions holds.
$f$2000 Mathematics Subject$Clas\mathit{8}ification$.

$37\mathrm{E}25,37\mathrm{B}20$.

Key wortls and phrases. transitive, strongly transitive, 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.

135(1) There exists a point with dense orbit.
(2) Whenever $U,$ $V$ are non-empty open sets, there exists an $n\geq 1$such that $f^{-n}(U)\cap V\neq\emptyset$ .
(3) The only closed invariant set $K$ with Int$K\neq\emptyset$

is $K=X$ .

Remark. We note that, in the case of a graph map : $Garrow G,$ istransitive if and only if for every pair of non-empty open sets andin , there exists a $k\geq 1$ such thatInt.
$f$

$f$

$U$

$G$

$U\cap$$V$$f^{k}(V)\neq\emptyset$

In the study of transitive maps, the subclass of those maps having all
iterates transitive plays a significant role. A map is totally transitiveifis transitive for all $n\geq 1$ (see [1]); note that a transitive map is
not always totally transitive.
A map : $Xarrow X$ is called strongly transitive if for every non-empty
open set of $X$ , there exists an such that.
We first call a useful proposition which shows a backward structureof a strongly transitive map for each point.
$f$$f^{n}$

$f$

$J$

$\bigcup_{k=0}^{n}f^{k}(J)=X$$n$

Proposition 2.1. Let : $Xarrow X$ be a map $ofX$ to itself. Then thefollowing are equivalent.
(1) For each.
$X,$(2) For every non-empty open set of.
$R_{4}rthermore$ ,if is open, then (1) and (2) are equivalent to(3) If
is a closed set with $f^{-1}(E)\subseteq E$ , thenor $X$ .
$f$$x \in X_{f}\mathrm{C}1\bigcup_{n=0}^{\infty}f^{-n}(x)=X$$U$$\bigcup_{n=0}^{\infty}f^{n}(U)=X$$f$$E=\emptyset$$E\subseteq X$

The examples below clarify the difference between transitivity andstrong transitivity properties.

Example 1. There exists a transitive map of the interval which is
not strongly transitive. This example appears in [3, Example 3] toillustrate another property. For completeness, we give a constructionof the map here.
Letbe a two-sided sequence of real numbers in $[0,1]$
such that$\{p_{n}|n\in \mathbb{Z}\}$,put$<p-2<p_{-1}<p_{0}<p_{1}<p_{2}<\cdots$and].
and $p_{-n}arrow 0$ when $narrow\infty$ . For
Define the map:
by $f_{n}(p_{n})=p_{n},$ $f_{n}(p_{n+1})=$, andis linear on the$[0,1]$ is given byintervals complementary to these points. : $[0,1]$$f(0)=0,$ $f(1)=1$ , and $f(x)=f_{n}(x)$ if $x\in$(see Figure 2 in [3]).
$p_{n}arrow 1$$n\in \mathbb{Z}$$f_{n}$$p_{n+1},$$I_{n}arrow I_{n-1}\cup I_{n}\cup I_{n+1}$$f_{n}( \frac{2p_{n}+p_{n+1}}{3})=p_{n+2},$$f_{n}( \frac{p_{n}+2p_{n+1}}{3})=p_{n-1}$$f$$f_{n}$—$I_{n}$$I_{n}=\lceil p_{n},p_{n+1}$

136
By Example 1 taken mod 1, we also have
Example 2. There exists a transitive map of the circle which is notstrongly transitive.

Letbe the bouquet with -petals generated by copies of theunit circle, where $n\geq 1$ . Using Example 1 taken mod 1 and a rotation
among petals with respect to the origin, we can easily have an exampleon$B_{n}$$n$$n$$B_{n}$.

Example 3. There exists a transitive map oftransitive.

$B_{n}$

which is not strongly

Example 4. Since the map in Example 1 is actually totally transitive
as stated in [3, Example 3], we have a totally transitive interval map
which is not strongly transitive. On the other hand, the interval map
below is strongly transitive, but not totally transitive. $g(x)=2x+$
$f$

$g$

$1/2,$ $(0\leqx\leq 1/4);-2x+3/2,$ $(1/4\leq x\leq 3/4);2x-3/2,$ $(3/4\leq x\leq$1).

3. MAIN RESULTS
Here is our main result.
Theorem 3.1. Let : $Garrow G$ be a graph map withfor each $k\geq 1$ . If is transitive, then it is strongly transitive.

$\#\mathrm{F}\mathrm{i}\mathrm{x}(f^{k})<\infty$

$f$

$f$

A map
$A$

in

$G$

$f$

is piecewise monotone if there is a finite set.
is monotone on each component ofon a graphsuch that

$f$

$G$

$G\backslash A$

Corollary 3.2. Let : $Garrow G$ be a piecewise monotone graph map.
If is transitive, then it is strongly transitive.
$f$

$f$

Remark. The interval case of the corollary above was proved by CovenMulvey [6].

Example 5. Let : $[0,1]arrow[0,1]$ be the map whose graph appears
is finite
below. Then is transitive and the set of fixed points offor each $k\geq 1$ . Therefore is strongly transitive, in fact, for each nondegenerate subinterval of $[0,1]$ , there exists an such that $f^{n}(J)=$$f$$f^{k}$

$f$

$f$

$J$

$[0,1]$

.

$n$

137
’

$,*|,| \int_{1}|l1\downarrow\{.|\mathfrak{l}|’’|\backslash$’$\iota_{\text{・},1}’$$\mathrm{t}’$’$\backslash _{l},\backslash$$\mathfrak{l}^{\backslash }\iota._{1}^{\backslash }\downarrow\mathrm{t}|\prime \mathrm{t}|$$_{l}’,,_{1}|’‘ i!_{\mathfrak{l}}1\iota’\prime\prime||$

’
${ }$

$\backslash \backslash \backslash$$\prime j\prime j|\prime\prime\prime’1|\mathrm{I}J|_{}’\text{・}\prime‘ \mathrm{I}\mathfrak{l}|,|,11\prime\prime|1’_{}||\mathfrak{i}\mathfrak{l},’$$j’,’‘’| \text{・}|,\prime\prime\prime\prime\prime\prime’_{}l\prime\prime_{1}\prime_{1}l_{}_{}\mathrm{t}\int_{\prime}^{\mathfrak{l}}’\iota,$

”

$,,$

$‘\backslash \mathfrak{i}^{\backslash }\mathfrak{l}^{\backslash }^{\backslash }_{l};1^{\backslash }|^{1}\prime 1\mathrm{J}’\backslash \uparrow|,!\mathfrak{l},|||$$\mathrm{t}\prime 1’\prime l\prime\prime||‘|,|\prime 1’,’\iota\iota,’,||\prime \mathrm{t},\backslash ’,$’$\prime 1\downarrow$’$|\prime 1_{l}\prime\prime_{l}$$\dot{r}|’,\uparrow\prime 1’,’\prime\prime||,|$’)1‘.’$1’\backslash .,\backslash \backslash ,$Proposition 3.3. Let : $Tarrow T$ be a totally transitive tree map.
Then is strongly transitive if and only if for evew non-degenerateconnected set of , there exists an $M$ such that for any $m\geq M$ ,$f$

$f$

$T$

$J$

$f^{m}(J)=T$.

The following generalizes the result for interval maps of CovenMulvey [6] to one for tree maps.
be theTheorem 3.4. Let : $Tarrow T$ be an onto tree map. Let
the least common
maximum order of any branch point in and. Then the following are equivalent.
multiple ofme to(1)is transitive and has a point of period which is.
2, . . . ,(2)is transitive.
(3) is totally transitive.
(4) is topologically mixing.
$\mathbb{R}rthermore$ ,is finite for each $k\geq 1$ , then the following
ifis equivalent to(5) for every non-degenerate connected set of , there exists anthat for any $m\geq M,$ $f^{m}(J)=T$ .
are well-known [8,Remark. The equivalences (1)Theorem 4.1], [1].
$\mathrm{v}(T)$

$f$

$T$

$N_{\mathrm{v}(T)}$$\{2, \ldots, \mathrm{v}(T)\}$$f$$pr\dot{\mathrm{v}}$$\mathrm{v}(T)$$f^{N_{\mathrm{v}(T)}}$

$f$

$f$

$\#\mathrm{F}\mathrm{i}\mathrm{x}(f^{k})$

$J$

$T$

$M\mathit{8}uch$$\Leftrightarrow(2)\Leftrightarrow(3)\Leftrightarrow(4)$4. REMARKS(I): It is useful to investigate the relation between the dynamics ofa graph map and the dynamics of the induced self-homeomorphism of
the inverse limit space [2], [3].
Let $f$ : $Xarrow X$ be an onto map. Associated with is the inverse$\ldots)|x_{i}\in X$ , and $f(x_{i+1})=x_{i}$ }, andlimit space (X, ){$f$$f$$=$$(x_{0},$ $x_{1},$138
the induced homeomorphism : (X, ) $arrow(X, f)$ (which is called theshift homeomorphism), given by $\hat{f}((x_{0}, x_{1}, \ldots))=(f(x_{0}), x_{0}, x_{1}, \ldots)$ .
Proposition 4.1. Let : $Xarrow X$ be an onto map of a metrizable
compact space X. If the shift homeomorphism : (X, ) $arrow(X, f)$ isstrongly transitive; then is strongly transitive.
$f$$\hat{f}$$f$$f$$\hat{f}$$f$

Unfortunately, the shift homeomorphism of a strongly transitivegraph map is not always strongly transitive. In fact, we have the following.
Proposition 4.2. Let be a non-degenerate graph and : $Garrow G$ bean onto map. Then the shift $h_{omeomo7}phismf$ : $(G, f)arrow(G, f)$ isstrongly transitive if and only if is the circle and is conjugate to
an imational rotation.
(II): We note that statement (2) in Proposition 2.1, which was introduced by Parry [7], implies strong transitivity for tree maps.
Proposition 4.3. Let : $Tarrow T$ be an onto tree map. Then isstrongly transitive if and only if for every non-empty open set of ,$G$

$f$

$G$

$f$

$f$

$f$

$U$

$\bigcup_{n=0}^{\infty}f^{n}(U)=T$$T$.

However, it is not always true for a general graph map.

Example 6. Let : $[0,1]arrow[0,1]$ be the map whose graph appears
bybelow. Using this map , we define the circle map :$0\leq\theta\leq
1$. Then the map is transitive and, wheresatisfies statement (2) in Proposition 2.1, but is not strongly transitive.
$f$

$f$

$g$$g(e^{2\pi i\theta})=e^{2\pi if(\theta)}$$S^{1}arrow S^{1}$$g$

REFERENCES[1] Ll. Alsed\‘a, M. A. del Rfo, and J. A. Rodr\’iguez, A splitting theorem for transitive maps, J. Math. Anal. Appl., 232(2), (1999), 359-375.

139[2] M. Barge and J. Martin, Chaos,odicity, and snakelike continua, Ttrans.
Amer. Math. Soc., 289(1), (1985), 355-365.
[3] M. Barge and J. Martin, Dense orbits on the interval, Michigan Math. J., 34(1), (1987), 3-11.
[4] A. M. Blokh, On transitive mappings of one-dimensional branched manifolds,
Differential-difference equations and problems of mathematical physics (Russian), Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 1984, pp.3-9, 131.
[5] A. M. Blokh, The connection between entropy and transitivity for onedimensional mapping8, Uspekhi Mat. Nauk, $42(5(257))$ , (1987), 209-210.
[6] E. M. Coven and I. Mulvey, Transitivity and the centre for maps of the circle,
Ergodic Theory Dynam. Systems, 6(1), (1986), 1-8.
[7] W. Parry, Symbolic dynamics and transformations of the unit interval, Trans.
Amer. Math. Soc., 122, (1966), 368-378.
[8] R. P. Roe, D ynamics and indecomposable inverse limit spaces of maps on finitegraphs, Topology Appl., 50(2), (1993), 117-128.
[9] R. P. Roe, Dense periodicity on finite trees, Topology Proc., 19, (1994), 237$pe7\dot{T}$248.

DEPARTMENT OF MATHEMATICS, SHIMANE UNIVERSITY, MATSUE, 690-8504,
JAPAN-mail address: yokoiQriko. shimane-u. ac. jp$E$