On positivity and universality of templates induced from diffeomorphisms of the disk (On Heegaard Splittings and Dehn surgeries of 3-manifolds, and topics related to them)

(1)

On positivity

and

universality

of templates

induced from diffeomorphisms of

the

disk

Mikami

Hirasawa・Gakushuin

Univ.

(

平澤美可三・学習院大学

)

$)$

Eiko Kin

1 ・

Kyoto

Univ.

(

金英子・

京都大学

)‘

1. INTRODUCTION

In thisnote, weconsider Inks induced ffom periodic orbits of orientation preserving

automorphisms $\varphi$of

$D^{2}$

.

We ffist present

some

basicterminologies. Wedenote the i-th

iteration of$\varphi$ by

$\varphi^{:}$

.

We say that $x\in D^{2}$ is aperiod $k\in \mathrm{N}$ periodic pointif$\varphi^{k}(x)=x$

and $\varphi^{:}(x)\neq x$for $1\leq i<k$

.

Inparticular, wesay that $x$is a

fixed

pointif$x$is aperiod 1periodic point. For $x\in D^{2}$, $\{\varphi^{:}(x)|i\in \mathrm{N}\}$ is called the orbit

of

$x$ and denoted by

$O_{\varphi}(x)$

.

If$x$ is aperiodic point, then $O_{\varphi}(x)$ is called the periodic orbit

of

$x$

.

Let $\Phi$ $=\{\varphi_{t}\}_{0\leq t\leq 1}$ be

an

isotopy of$D^{2}$ such that $\varphi_{0}=id_{D^{2}}$, $\varphi_{1}=\varphi$

.

For afinite

union of periodic orbits $P$ of$\varphi$,

we

define asubset of$\tilde{V}=D^{2}\mathrm{x}S^{1}(\cong D^{2}\mathrm{x} I/(x, 0)\sim$

$(x, 1))$, denoted by $S_{\Phi}P$,

as

follows.

$S_{\Phi}P= \bigcup_{0\leq t\leq 1}(\varphi_{t}(P)\mathrm{x}\{t\})/(x,0)\sim(x, 1)$

.

$S_{\Phi}P$is called a suspension

of

$P$ by $\Phi$

.

Let $V$ be astandardly embeddedsolid torus in

the $3\underline{- \mathrm{s}}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}S^{3}$

.

Then$h:Varrow V$ denotes ahomeomorphism such that for alongitude

$\tilde{\ell}$

on

$V$, $h(\tilde{\ell})$ is aknot with the lnking number of$h(\tilde{\ell})$ and the

core

circle of_$V$ being 1

(see Figure 1). For each $i\in \mathrm{Z}$, $h^{:}(S_{\Phi}P)$ is alink in $S^{3}$, where the orientation of $S_{\Phi}P$

is induced from parametrization by $t$

.

$\tilde{V}$

$V$

Figure 1

Partiallysupported by JSPS ResearchFellowshipsfor YoungScientist$\mathrm{s}$

数理解析研究所講究録 1229 巻 2001 年 146-150

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On positivityand universalityoftmplatae inducedffomdiffeomorphismsof the disk

Definition 1.1. Let $\varphi$ :

$D^{2}arrow D^{2}$ be

an

orientation preserving automorphism, and

$\Phi$ $=\{\varphi_{t}\}_{0\leq t\leq 1}$

an

isotopy of $D^{2}$ such that $\varphi_{0}=id_{D^{2}}$, $\varphi_{1}=\varphi$

.

We say that $\varphi$ induces

all link types if there exists

an

integer $i\in \mathrm{Z}$ satisfyingthe followingconditions.

$(*)$ For each link $L$ in $S^{3}$, there exists afinite union of periodic orbits $P_{L}$ of

$\varphi$ such that $L=h^{:}(S_{\Phi}P_{L})$

.

We note that the definitiondoes not depend

on

$\Phi$

.

Moreover the number of integers

$i$ suchthat $h^{i}$satisfies

$(*)$ does not depend

on

$\Phi$ (see [8]). Hencewedenote the number

by $\overline{N}(\varphi)$, that is,

$\overline{N}(\varphi)=\#$

{

$i\in \mathrm{Z}|i$ satisfies $(*)$ for $\Phi$

}.

The topological entropy $h_{t\varphi}(\varphi)$ for $\varphi$ is

ameasure

ofits dynamicalcomplexity (see

[14] for adefinition of the entropy). Aresult of GambaudO-van

Strien-Tresser

([3, Theorem $\mathrm{A}$

]) tells us that if $h_{t\varphi}(\varphi)=0$, then $\varphi$ does not induce all link types, i.e.,

$\overline{N}(\varphi)=0$

.

Itis natural to ask the following problem:

Problem 1.2. Which automorphism induces all link types ?

In [11], the second author researched the Smale horseshoe map [13] on Problem 1.2.

The Smale horseshoe map is afundamental example to study complicated dynamics

since the invariant set is hyperbolic and is conjugate to the 2-shift, and such invariant

sets are often observed in many dynamical systems [9] (see [12] for basicdefinitions of

dynamical systems).

Theorem 1.3. [11] Let $H$ be the Smale horseshoe map. Then $\overline{N}(H)=\overline{N}(H^{2})=0$

and$\overline{N}(H^{3})=1$.

Since $h_{top}(H)$ and $h_{\hslash\varphi}(H^{2})$

are

positive, Theorem 1.3 shows the existence of

diffe0-morphisms not inducing all link types.

We will consider Problem 1.2 forgeneralizedhorseshoe maps$G$ using twist signature

$t(G)$ (see Definitions 2.1, 2.2). In Theorem 3.1, we completely _determine _{the number}

$\overline{N}(G)$ by $t(G)$

.

2. GENERALIZED HORSESHOE MAP AND TWIST SIGNATURE

For definitions ofgeneralzed horseshoe map and twist signature,

we

first introduce

some

terminologies. Let $R=[- \frac{1}{2}, \frac{1}{2}]\mathrm{x}[-\frac{1}{2}, \frac{1}{2}]\subset D^{2}$, and let 50, $S_{1}$ be half disks

as

in Figure $2(\mathrm{a})$

.

For $c,d$ $\in[-\frac{1}{2}, \frac{1}{2}]$, we $\mathrm{c}\mathrm{a}\mathrm{U}$ _{$\ell_{v}=\{c\}\mathrm{x}[-\frac{1}{2}, \frac{1}{2}]$} (resp.

$\ell_{h}=[-\frac{1}{2},$ $\frac{1}{2}]\mathrm{x}\{c’\}$) $a$

vertical (resp. a horizontal) line. For $[c, d]$, $[d, d’] \subset[-\frac{1}{2}, \frac{1}{2}]$,

we

call $B=[c,d] \mathrm{x}[-\frac{1}{2}, \frac{1}{2}]$

(resp. $B’=[- \frac{1}{2}\}\frac{1}{2}]\mathrm{x}[c’,$$d’]$) a vertical (resp. a horizontal) rectangle.

Let $B_{1}$, $B_{2}$ (resp. $B_{1}’$, $B_{2}’$) be disjoint vertical (resp. disjoint horizontal) rectangles.

The notation $B_{1}<_{1}B_{2}$ (resp. $B_{1}’<_{2}B_{2}’$)

means

the first (resp. second) coordinate of

apoint in $B_{2}$ (resp. $B_{2}’$) is greater than that of $B_{1}$ (resp. $B_{1}’$). We denote the open

rectangle which lies between $B_{1}$ and $B_{2}$ by $(B_{1}, B_{2})$

.

Definition 2.1. Let $n\geq$

.

2 be

an

integer. A generalized horseshoe map $G$

of

length_$n$

is

an

orientation preservingdiffeomorphism of$D^{2}$ satisfying the following: There exist

vertical rectangles $B_{1}<_{1}B_{2}<_{1}\cdots<_{1}B_{n}$ and horizontal rectangles _{$B_{1}’<_{2}B_{2}’<_{2}$}

$\ldots<_{2}B_{n}’$ such that

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M. Hirasawa&E. Kin

(1) for each $1\leq i\leq n$, $G(B_{\dot{1}})=B_{j}’$ for

some

$1\leq j\leq n$,

(2) for each $1\leq i\leq n-1$, $G((B\dot{.}, B_{i+1}))\subset S_{k}$ for

some

$k\in\{0,1\}$,

(3) $G$expandsthe part of horizontal lines which intersects each$B_{i}$ uniformly,

and contract the vertical lines in each $B_{i}$ uniformly, (4) $G|_{S\mathrm{o}}$ : $S_{0}arrow S_{0}$ is contractive,

(5) if$n$ is even (resp. odd), then $G(S_{1})\subset Int$ $S_{0}$ (resp. $G|s_{1}$ : $S_{1}arrow S_{1}$ is

contractive) and

(6) $G$ has

no

periodic points in $D^{2}\backslash R$

.

Definition 2.2. Let $G$ be ageneralized horseshoe map of length $n$

.

Twist signature

$t(G)$

of

$G$ is the arrayof$n$ integers $(a_{1}, \cdots, a_{n})$ satisfying the following:

(1) $a_{1}=0$

.

(2) For $2\leq i\leq n$, $a:=a_{\dot{|}-1}+1$ if $G(B:-1)<_{2}G(B_{\dot{1}})$ and $G((B:-1, Bi))\subset$

$S_{1}$, or if $G(B_{\dot{l}}-1)>_{2}G(B_{\dot{1}})$ and $G((B_{\dot{|}-1}, B:))\subset S_{0}$

.

Otherwise

$a_{i}=\alpha_{-1}.-1$

.

By the condition of generalzed horseshoe maps $G$,

$\mathrm{A}=\cap G^{m}(B_{1}m\in \mathrm{Z}\cup\cdots\cup B_{n})$ is

hyperbolic which is conjugate to the n-shift.

Notice that the Smale horseshoe map is ageneralized horseshoe map of length 2

with twist signature $(0, 1)$

.

(a)

$\mathrm{t}^{\epsilon})$ (d)

Figure 2

Example 2.3. (1) Let $K_{1}$ be ageneralized horseshoe map of length 3as in Figure

$2(\mathrm{b})$

.

Then $t(K_{1})=(0,1,2)$

.

(2) Let $K_{2}$ be ageneralized horseshoemapof length 4as in Figure$2(\mathrm{c})$

.

Then$t(K_{2})=$

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Onpositivityanduniversalityoftemplat einducedkomdiffeomorphismsof the$\mathrm{d}_{\dot{\mathrm{B}}}\mathrm{k}$

$(0, 1, 0,$$-1)$

.

(3) Let $K_{3}$ be ageneralizedhorseshoe mapof length4as in Figure$2(\mathrm{d})$

.

Then$t(K_{3})=$

$(0, -1, -2, -3)$

.

3. STATEMENT OF RESULTS

Let $G$ be ageneralized horseshoe map with twist signature $(a_{1}, \cdots,a_{n})$

.

We say

that $G$ is positive (resp. negative) if forany _{$i\in\{1, \cdots,n\}$}, $a_{t}\geq 0$ (resp. $a:\leq 0$). We

say that $G$ is mixedif$G$ is neither positive

nor

negative. For example, $K_{1}$,_{$K_{2},K_{3}$} in

Example 2.3

are

positive, mixed, negative respectively.

The following is Main theorem ofthis note:

Theorem 3.1. For$x\in \mathrm{R}$, let_$[]$ be thegreatestintegerwhich does not exceed_$x$

.

Let$G$

be ageneralizedhorseshoe map withtwist signa

rure

$(a_{1}, \cdots,a_{n})$

.

Let_{$M_{+}= \max\{a:|1\leq$}

$i \leq n\}andM_{-}=\dot{\mathrm{m}}\mathrm{n}\{a_{i}i\leq n\}.IfGnegative_{f}then\overline{N}(G)=[\frac{-M_{-}-1|1\leq}{2}].IfGism$$\dot{a}ed,ihen^{\frac{ve}{N’}}(G)=[\frac{\overline{N}(G)M+-1}{2}]+]+1ispositithen=[\frac{M_{+}-1}{[\frac{-M_{-}-12]}{2}}.IfG$

.

is

The next corollary is adirect consequence of the above theorem:

Corollary 3.2. Let $G$ be a generalized horseshoe map, and _{$M_{+}$} and $M_{-}$ be as in

Theorem 3.1. Then $G$ induces all link types, _$i.e.$, $\overline{N}(G)\geq 1$

if

and only

if

$G$ _is one

of

thefollowing types.

$\bullet$ $G$ is positive and $M_{+}\geq 3$

.

$\bullet$ $G$ is negative and$M_{-}\leq-3$

.

$\bullet$ $G$ is mixed.

Recall that $K_{1}$,$K_{2}$,$K_{3}$

are

generalized horseshoemapsin Example

2.3.

ByTheorem

3.1, $\overline{N}(K_{1})=0$, $\overline{N}(K_{2})--\sim 1$ and $\overline{N}(K_{3})=1$

.

The proof ofTheorem 3.1 is done byusing the template theory ([2], [4], [5]).

REFERENCES

[1] J. Birman, Braids, Links, and Mapping Class Groups, Ann. Math. Studies 82, PrincetonUniv.

Press, Princeton (1974).

[2] J. Birman and _{R. Williams, Knotted}_{periodic orbits in} _dynamical _systems-H: _{Knot holders}_for

fiberedknots, Cont. Math. 20 (1) (1983) 1-60.

[3] J. Gambaudo,S.vanStrienandC.$?$}$\varpi \mathrm{e}\mathrm{r}$, The$per\cdot M^{\cdot}c$orbitsstructure oforientationpreserving

diffeomo\prime phf.sms on $D^{2}$

with $topolo\dot{\varphi}cal$ entropy zero, Ann. Inst. H. Poincar _\’e

Phys. Theor. 50

(1989) 335-356.

[4] R. Ghrist, Branched _{twO-manifolds} supportingall links, Topology 36 (2) (1997) 423-448.

[5] R. Ghrist, P. Holmes,andM. Sullivan, Knots and LinksinThreeDimensionalFlows, Lect. Notes

inMath. 1654, Springer-Verlag.

[6] R. Ghrist and T. Young, From Morse-Smale to allknots andlinks, Nonlinearity 11 (1998)

1111-1125.

[7] V. Hansen, Braidsand_{coverings: selected topics, London Math. Soc. Stud.} _Texts _18,_Cambridge

University Press (1989).

[8] M. Hirasawaand E. Kin, Onpositivity and universality _of_{templates associated with generalized}

horseshoe maps _ofthe disk, inpreparation.

[9] A. _{Katok, Lyapunov exponents, entropy and} _{periodic orbits}_for _{diffeomorphisms,} _{Publ. Math.}

IHES 51 (1980) 137-174

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M.Ih.raeawa&E. Kin

[10] E. Kin, A suspension _ofan orientation preserving diffeomorphism _of$D^{2}$ wih a hyper

point and universal template, J. Knot Thy. andRam. 9(6) (2000) 771-795.

[11] E. Kin, The thirdpower_ofthe Smale horseshoe induces all link types, J. Knot Thy. al (7) (2000) 939-953.

[12] C. Robinson, Dynamical Systems, Stability, Symbolic Dynamics, and Chaos (seconc

CRC Press, Ann Arbor, M (1995).

[13] S. Smale,

_{Differentiable}

dynamical sytttevna, Bull, Am. Math. Soc., 73 (1967) 747-817.

[14] P. Walters, Anintroduction to ergodic theory, Springer(1982).

Department of Mathematics, Faculty of Science, Gakushuin University,

1-5-1

Mejiro Toshima-ku, Tokyo 171-8588Japan

$E$-mail address $\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\epsilon \mathrm{a}\mathrm{w}\mathrm{a}\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{h}$.gakushuin.ac.jp

Department ofMathematics, Kyoto University,

Oiwakecho Kitashirakawa SakyO-ku KyotO-shi, Kyoto $\infty$-8502Japan

$E$-mail address kinOkusm.$\mathrm{k}\mathrm{y}\mathrm{o}\mathrm{t}\mathrm{o}-\mathrm{u}.\mathrm{a}\mathrm{c}$.jp