数理解析研究所講究録
1118 巻 1999 年 67-83

67

One-dimensional three-body problem
via symbolic dynamics
Kiyotaka Tanikawa1 and Seppo$\mathrm{M}\mathrm{i}\mathrm{k}\mathrm{k}_{0}1\mathrm{a}^{1,2}$

谷川清隆、 セッポミッコラ

Astronomical Observatory, Mitaka, Tokyo 181, Japan
Observatory, University of Rrku, 21500 Piikki\"o, Finland$\perp \mathrm{N}\mathrm{a}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{a}1$$2\mathrm{R}\mathrm{o}\mathrm{r}\mathrm{l}\mathrm{a}$Abstract
Synbolic dynamics is applied to the one-dimensional three-body problem. The sequence of collisions along an orbit is expressed as a symbol
sequence of three symbols. We find systematically sequences of collisions
which are unallowable. There is an infinite number of periodic sequences
which suggets an infinity of periodic orbits other than the Schubart orbit.
Under reasonable assumptions on unallowable sequences, we prove thatsymbol sequences form a Cantor set.

Key words: one dimensional three-body problem–symbolic dynamics–chaos

1

Introduction

One-dimensional three-body systems starting from general initial conditions havebeen extensively studied by Mikkola and Hietarinta(1989; 1990; 1991). Tanikawa&Mikkola(1999; hereafter referred to as Paper I) introduced symbol sequences
and found that triple-collision orbits can be obtained easily as boundaries ofdifferent symbol sequences. (For the application of symbol sequences to the threebody problem, see also Zare&Chesley, 1998). They found that the region of thephase space so far considered chaotic is stratified by regions separated by triplecollision curves, i.e., curves formed by initial conditions leading to triple collision.
The present paper is the extension of Paper I. Our purpose here is to understand more deeply the structure of the phase space of the one-dimensional
three-body problem and to get an insight into its dynamics. Instead of considering orbits themselves, we consider the set of symbol sequences constructed fromthe sequences of binary collisions along orbits. Our main results are as follows.
We divide the surface of section into five regions with different types of symbol

68sequence and use the symmetry of the surface of section. Then we find that there
are unallowable sequences of collisions. Next, we find that the surface of section israther simply stratified by bands of points with symbol sequences with increasingor decreasing order when con.sidered as decimal numbers. Based on this fact, theexistence of periodic orbits and other orbits are shown. Finally, based on the
numerical results, we prove that the set of allowable symbol sequences form a
Cantor set in the set of all symbol sequences.

2

Formulation of the Problem, and $m_{2}(m_{0}=m1=m_{2})$ in this order on a line.
We put three mass points
Fix the masses and the gravitational constant to one. Then the Hamiltonian of
the problem is given (1, Mikkola&Hietarinta, 1989) by$m_{1},$ $m_{0}$$H= \frac{1}{2}\sum_{i=0}w_{i}22-\sum_{i<j}\frac{1}{|_{X_{i^{-}}}x_{j}|}$where$w_{i}$are momenta conjugate to the coordinates. Introducing new coordinates by

.

(2.1)

on the line. We have$x_{i}$$x_{1}\leq x_{0}\leq x_{2}$$q_{1}=x_{0}-X1$,$q_{2}=x_{2}-x0$

,

(2.2)

we get the new Hamiltonian as$H=p_{1}^{2}+p_{2^{-}}^{2}p_{1p_{2}}- \frac{1}{q_{1}}-\frac{1}{q_{2}}-\frac{1}{q_{1}+q_{2}}$

.

(2.3)

We fix the total energy to-l and start the integration at $q_{1}(0)=q_{2}(0)=R$ .
This means that two outer particles are placed in an equal distance from thecentral. Then the value of potential is fixed to 2. $5/R$ . The kinetic energy isby$T=2.5/R-1$ .
(2.4)
$T$

d.etermined

Initial zero velocities correspond to $R=1/2.5$ .
If we introduce a parametrization

then$(\theta, R)$$\sqrt{3}(p_{1}-p2)=$$2\sqrt{T}\sin\theta$,$(p_{1}+p2)=$$2\sqrt{T}\cos\theta$

,

(2.5)

specify the initial value. Velocities are given by$\dot{q}_{1}=2p_{1^{-}}p_{2}=2\sqrt{T}\cos(\theta-\pi/3)$$\dot{q}_{2}=2p_{2^{-}}p_{1}=2\sqrt{T}\cos(\theta+\pi/3)$

.

$R$

and

$\theta$

as
(2.6)

69
The function $S=p_{1}Q^{2}1+p_{2}Q_{2}^{2}$ generates the additional transformation with the
new coordinates
and new momenta $P_{i}=2Q_{i}p_{i}$ . If accompanied with the time transformation $t’=q1q2$ the regularized new Hamiltonian$Q_{1}=\sqrt{q_{1}},$$Q_{2}=\sqrt{q_{2}}$$\Gamma=\frac{1}{4}(P_{1}^{2}Q^{2}2+P_{2}^{2}Q21^{-}P1P2Q_{1}Q2)-Q_{1}^{2}-Q_{2}^{2}-Q12Q^{2}2/(Q_{1}2Q_{2}+)2-Q2Q_{2}21E,$$(1)$

is obtained. Here $E$ is the (initial) numerical value of the Hamiltonian. The
equations of motion following from this Hamiltonian can be integrated numerically with conventional methods, such as the Bulirsch-Stoer integrator (Bulirsch&Stoer 1966).
We start integration from the state $q_{1}=q_{2}$ with $0\leq\theta<\pi$ . For our fixedenergy, initial states can be expressed in a surface of section $H$ :$H=\{(\theta, R)|0\leq\theta<\pi, 0\leq R\leq 2.5\}$.

To obtain the global structure, we cover surface of section $H$ with a mesh ofgrid size
and integrate orbits starting at each verticesof grids forward until the 66th binary collision is obtained. The total number oforbits amounts to $2.25\cross 10^{6}$ . To obtain particular local structure, we perform
additional integrations with finer mesh.
$(\triangle R, \triangle\theta)=$

3$(0.002, 0^{\mathrm{o}}.1)$

Symbol Sequences and Surface of Section

There can be three types of collision along an orbit: binary collisions between
andand between particles
andand a triple collision. Let us denote abinary collision between
by ’1’ and a binary collision
and
by
and
’2’, and a triple collision by ’0’. A symbol sequence is constructed in such a waythat whencollide, the symbol ’1’ is added to the sequence and whenandandcollide the symbol ’2’ is concatenated.
Let us express an orbit as a sequence of $0,1$ and 2 as follows:$m_{1}$$m_{0}$$m_{0}$$m_{0}$$m_{0}$$m_{2}$$m_{2}$$m_{1}$$m_{2}$$m_{0}$$m_{1}$$m_{0}$$(. . . n_{-2}n_{-1}.n_{0}n_{1}n2\cdots)$$i\neq 0$ are either $0,1$ , or 2.
where
We follow the orbit starting from the initial condition defined in \S 2 to the
future and to the past. Then
represents the first collision.
repreandsent the type symbol of the second and third next binary collisions, and so on.
Similarly,
represent the type symbol of the past collisions.
Let denote the set of all -infinite sequences $s=$ $(. . . n_{-2}n_{-1}.n0n1n_{2}\ldots)$ .
We define a metric on setting $\mathrm{d}(s, s)=0$ andif
and$n_{i},$$i\in \mathrm{Z},$$n_{0}$$n_{-1},$ $n_{-2},$$n_{1}$$n_{2}$$\ldots$$\Sigma$$\mathrm{b}\mathrm{i}$$\Sigma$$\mathrm{d}(s^{12}, s)=3^{-|m|}$$s^{1}\neq s^{2}$

70

is the least integer such thatcompact metric space. Let us define$|m|$$s_{m}^{1}\neq s_{m}^{2}$$\Sigma_{2}=$(Block&Coppel, 1992). Then$\Sigma$is a{ $s\in\Sigma|s_{i}=1$ or 2}.

Then it is known thatis a Cantor set(Robinson, 1995). In other words, symbol
sequences corresponding to orbits which repeat binary collisions form a Cantor
set in the whole sequence space .
The shift operator is defined by$\Sigma_{2}$$\Sigma$$\sigma$$\sigma(\ldots n_{-2}n-1\cdot n0n1n_{2}\ldots)=$on$\Sigma$$(. . . n_{-2}n_{-1}n0\cdot n_{1}n_{2}\ldots)$.

Remark. We cannot directly obtain orbits which startend at triple
collision. In our numerical study, though we assigned symbol ’0’ to triple collision,
we actually work in the space . Symbol sequences containing ’0’ can be obtained
only as a boundary of symbol sequences not containing ’0’. This has been themain result of paper I.
$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\Sigma_{2}$The correspondence between a symbol in a sequence and a point in the surfaceof section is not. It follows from the following propositions that if theorbit appears again on the surface $H$ , then the preceding and succeeding binary
collisions occur withand, respectively. To put it
and withanddifferently, the period(present) can only be inserted between successive ’1’ and ’2’in this order. An orbit does not appear on $H$ as long as binary collisions between
the same particles are repeated.
$\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{t}_{\mathrm{o}^{-}\mathrm{o}\mathrm{n}\mathrm{e}}$$m_{1}$$m_{0}$$m_{2}$$m_{0}$

Proposition 3.1. A trajectory in the-plane crosses transversely the homographic line except at $(\theta, R)=(0,0)$ , if it does at all.
$(q_{1}, q_{2})$Proof. If, the solution is homographic and always on theandhomographic line. So the trajectory initially outside the homographic line
become tangent to the line except the case of triple collision.
$q_{1}=q_{2}$$\dot{q}_{1}=\dot{q}_{2}$$\mathrm{n}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\square$$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}_{0}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\backslash 3.2$

. If a trajectory crosses the homographic line on the$(q_{1}, q_{2})-$plane, a binary collision occurs before the trajectory again crosses it.

-side
Proof. Suppose that a trajectory crosses the homographic line from, i.e.,
to-side at
andat
. Then there exists a$(q_{1}<q_{2})$$(q_{1}>q_{2})$$\mathrm{t}=t_{0}$$q_{1}=q_{2}$$\dot{q}_{1}>\dot{q}_{2}$$t=t_{0}$smallsuch that $q_{1}>q_{2}$ andat $t=t_{0}+\epsilon$ . This implies that there
needs a finite time in order to again cross the homographic line. Suppose thatthe first crossing occurs at
without binary collision during. We
can use the continuity argument in this case. At $t=t^{*}$ , we should have $q_{1}=q_{2}$by Proposition 3.1. In order to have this, we should have $q_{1}>q_{2}$and$\epsilon>0$$\dot{q}_{1}>\dot{q}_{2}$$t=t^{*}>\mathrm{t}_{0}$$\dot{q}_{1}<\dot{q}_{2}$$(t_{0}, t^{*})$

71

Figure 1: Stratified structure of the surface of section. Reproduction of Fig.5
in Paper I. Shown are the Schubart region(black), triple collision curves, andimmediate escape reions(blank).
$t_{0}<t^{**}<t^{*}$ . We should have in turnat someat least at
some time between and . This implies $q_{2}<q_{1}$ at the same instant, whicha contradiciton.

and$\dot{q}_{1}=\dot{q}_{2}$$t^{**},$$t_{0}$$\ddot{q}_{2}>\ddot{q}_{1}$$t^{**}$$\square \mathrm{i}\mathrm{s}$Finally, the surface of section has a symmetry due to the reversibility of the
problem. The past orbit starting at pointis realized by the future orbit
starting at point $(\pi-\theta, R)$ with particle names
andexchanged. Thus, to$<t<\infty$
theobtainorbits for-ooof all the points of the surface of section, we
need only to integrate their orbits for $0\leq t<\infty$ .
$(\theta, R)$$m_{1}$$m_{2}$

4

Orbits and Symbol sequences

4.1

Global structure of the Surface of Section

In paper I, we observed that the surface of section $H$ is divided into smallerparts by points with different symbol sequences. In fact, $H$ is divided into two
regions of points having symbol sequence 1.22. . . and 1.21. ... It is divided into
four by symbol sequence 1.222. .., 1.221. . ., 1.211.. ., and 1.212. . .. It is divided
${ }$

72

.

into seven by sequences 1.2222. . ., 1.2221. . ., 1.2212. . ., 1.2122. . ., 1.2121. .,1.2112..., and 1.2111. . .. An so on. The boundary of regions of different symbol
sequences form curves and this curve turned out to be initial conditions of triple
collision. We called these curves triple collision curves. Thus the surface ofsection is divided into increasing number of smaller parts with increasing numberof digits of symbol sequences. (see Fig.1, a reproduction of Fig.5 in paper I.)
In paper I, we did not consider the organization or distribution of various
regions with respect to different symbol sequences. In the present paper, ourtarget is to know the detailed structure of the surface of section and get an
insight into the dynamics of the one-dimensional three-body problem. The firstimportant observation is that there is a large structure in $H$ . Four curves seem toemanate from the corner of the black region, i.e. the ’Schubart region’ surrounding
the stable periodic orbit called the Schubart orbit, and reach the bottom boundaryof the surface of section. In Fig. 1, these four curves are obtained as a subset of
scattered orbits starting just outside the Schubart region.
In order to confirm that these are actually curves, we performed orbit intergrations with initial conditions along short segments across these supposed
for horizontal search, and it is 0.0001
curves. The grid size is as small as
for vertical search. A point is considered to be on the curves if the corresponding
symbol sequence is of the form $1.(21)^{n}$ with $n\geq 36$ . Near the boundary of the
Schubart region, more digits are required. Moreover, there we interpolated theposition of the curve when more than one grid points satisfy the condition.
. Four curves seem smoothly connected to
The result ,is shown in Fig.
the boundary of the Schubart region. We denote the regions separated by these, and , meaning the Schubart region, left, center, right,
curves by $S,$
and top regions, respectively. The corresponding symbol sequences are given in
Table I. Here, for example, (21) means 2121. . . 21 ( times).
$0^{\mathrm{o}}.001$$2(\mathrm{a})$$\mathcal{L},$$C,$$\mathcal{T}$$\mathcal{R}$$n$Table I. Symbol sequences in the surface of section.

Region Sequence $(n\geq 0)$(21)$\backslash$$S$$\mathcal{L}$$C$$\mathcal{R}$$T$$.(21)^{\infty}$... $1.(21)^{2}n22\cdots$. .. $1.(21)^{21}n+1\cdots$$\ldots 1.(21)^{21}n+22\cdots$$\ldots 1.(21)^{22}n+1\cdots$, and is based on the numerical result
Remark. The division into $S,$that triple collision curves do not cross the boundaries of these regions. One
may not be exact near theshould be careful that the division seen in Fig.
boundary of . There is a pair of period-2 points around the Schubart region.
$\mathcal{L},$$C,$$\mathcal{R}$$\mathcal{T}$$2(\mathrm{a})$

$S$

73

$3\mathrm{U}$$3\mathrm{U}$100$1\mathrm{O}\mathrm{O}$

theto
theta

theto

150

Figure 2: (a) Division of the surface of section. (b) Overlapping regions of thesurface of section and its reverse.

These points at the corners of the Schubart region and the outermost KAM curveof the Schubart region are expected disjoint. Near the boundary of the Schubart, and $T$ may be intermingled in a complicated manner. In thisregion,paper, we will not draw any conclusion related to this area.
$\mathcal{L},$$C,$$\mathcal{R}$

74

4.2

Unallowable symbol sequences

As we pointed out in \S 3, the surface of section has a symmetry. The past history
is realized by the future of the orbit starting atof the orbit starting at$(\pi-\theta, R)$ and vice versa. Thus, in order to get the-infinite symbol sequence
we calculate the future orbit starting atof an orbit starting at
sequence
1.2
calculate the future orbit
and record its future symbol$(\pi-\theta,R)$
makestarting at
and record its future symbol sequence
$s=2$
where
or 1, andor 2 according asa past symbol sequence.. .
.
(iv) concatenate two symbol sequences as...
The above procedure suggests us to check whether or not a given symbol
sequence is $all_{ow}able$ ( , realizable as a sequence of collisions). If an arbirary future symbol sequence can be concatenated with an arbitray past symbol sequence,then any -infinite symbol sequence is allowable.
and callLet us reverse the surface of section $H$ with transformation
the reversed surface of section. Let us denote respectively
the resultant surface$(\theta, R)$$\mathrm{b}\mathrm{i}$$(\theta, R),$$(\theta, R)$$(\mathrm{i})$$c_{2}c_{3}\ldots,$$(\mathrm{i}\mathrm{i})$$1.2s_{2}s_{3}\ldots,$$\overline{s}_{3}\overline{s}_{2}1.2$$(\mathrm{i}\mathrm{i}\mathrm{i})$$\overline{s}=1$$\overline{s}_{3}\overline{s}_{2}1.2C_{2^{C_{3}}}\ldots$$\mathrm{i}.\mathrm{e}.$$\mathrm{b}\mathrm{i}$$\thetaarrow\pi-\theta$$H^{t}$$C’,$
the regions of . Here for example,
is the original if notbyandreversed. The corresponding sequences (for the past) with ’1’ and ’2’ exchanged
are given in Table II.
$\mathcal{L}’,$$\mathcal{R}’$$H^{t}$$\tau/$$\mathcal{L}’$$\mathcal{R}$

Table II. Symbol sequences in the reversed surface of section.
$\frac{\mathrm{R}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{y}\mathrm{m}\mathrm{b}\mathrm{o}1\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}(n\geq 0)}{\mathcal{L}’\ldots 11(21)2n+\mathrm{l}.2}\ldots$$C’$$\mathcal{R}’$$\mathcal{T}’$. .. $2(21)^{21}n+.2\cdots$... 11 (21)$.2\cdots$.. , $2(21)^{22}n+.2\cdots$

If regions without prime and with prime overlap, then the symbol sequenceobtained by concatenating two one-sided sequence is allowable. Otherwise thesequence is unallowable. There are 17 combinations of possible overlapping re$T$
gions: 16 combinations between one of $L,$
and one of $L’,$
and one$S’$
combination of and . Due to the symmetry of the problem, we have $S=S’$ .
. As we pointed out inWe show the non-empty overlapping regions in Fig.
$S\capS’$
Remark in \S 4.1, near the boundary ofother overlappng regions may havecomplicated form. However, we neglect it.
Non-overlapping combinations of regions are listed in Table III together withunallowable words, where a word means a finite sequence of symbols. Any symbol
sequence having an unallowable word as its subsequence is unallowable. This canbe easily shown by using the shift operator .
$C,$$c^{J},$$\mathcal{R},$$S$$2(\mathrm{b})$$\sigma$$\mathcal{R}/,$$\tau/$

75

Table III. Pairs of non-overlapping regions and unallowable words.

There are no other unallowable words. In fact, the regions, and $T$
are stratified by triple collision curves running almost parallel to their boundary
inside the surface. In addition, all these stratified regions in, andconverge to triple collision points at the bottom of the surface of section. On
the other hand, triple collision curves of the original and reversed surface ofsections intersect transversely. This can be confirmed comparing Figs. 1 and 3.
$C’,$ $R’$ , andTherefore, if one of, andhas common area with one of$T’$ , then the region for any sequence...
in the former and the region forany sequence. in the latter has common points, and the concatenated$(.
.
.
S-2S_{-}1\cdot
t0t1\cdots)$sequence
is allowable.
$\mathcal{L},$$\mathcal{L},$$\mathcal{L},$$C,$$\mathcal{R}$

-.

$C,$

$\mathcal{R}$$\mathcal{R}$$\mathcal{T}$$\mathcal{T}$$\mathcal{L}’,$$($$(.t_{0}t_{1}$$C,$$s_{-21}s_{-}.)$$)$

Proposition 4.1. Unallowable words have the form $2(21)^{2k}+122,11(21)^{2}k22$ ,11 21 $)^{2k}+11$ , and $2(21)2k+31$ for $0\leq k\leq k_{0}$ . Hereis some large positive$4k$
integer. Their lengths are
and $4k+1$ with $k\geq 1$ .
$($$k_{0}$

We give first several examples in Table IV. The absense of These sequences
are confirmed numerically.
Table IV. Unallowable words of length less than or equl to 13.

Remark. In view of Remark in 4.1, we added the restriction
tion 4.1.

4.3$k\leq k_{0}$in Proposi-

A detailed structure of the surface of section

In order to see the detailed distribution of symbol sequences in the surface ofsection, we integrated orbits along the lines, and 150 with a finer$\theta=30^{\mathrm{O}},$$90^{\mathrm{O}}$

76

100

50

$,$

theta
theto

$\sim\sim$

Figure 3: A finer structure of the surface of section.

mesh. We find that along the line
which is transeverse to the structure
in , symbol sequences are distributed in such a way that symbol sequences
considered as ternary or decimal numbers decrease with increase of starting. We divide symbol sequences into three groups:
fromdown to$1.(2)^{n}1\ldots$ , and $1.(21)^{2}k22\ldots$ with $n\geq 2,$ $k>0$ . We call the regions with, and , respectively. The lack of odd exponents inthese symbol sequences
the symbol sequences inwill be understood if one sees the symbol sequences
in introduced below.
Along the line
and in , symbol sequences are distributed in such a waythat symbol sequences considered as ternary or decimal numbers increase with
up to
increase of starting from. We divide symbol sequences$1.2(1)^{n}2\ldots$ , and $1.(21)2k+11\ldots$ with $n\geq 2,$ $k>0$ . We
into three groups:
call the regions with these symbol sequences, and , respectively. Similar
to the case of , the lack of even exponents in the symbol sequences in will be
understood if one sees the symbol sequences in introduced below.
Along the line
and in , symbol sequences decrease with increaseof. We divide symbol sequence into threefromdown to$1.21(2)^{n}1.*\cdot$ , and $1.(21)^{2k+1}22\ldots$ with $n\geq 2,$ $k>0$ . We callgroups:, and , respectively.
the regions with these symbol sequences, oralong either of the lines
and in , symbol$\theta=30^{\mathrm{O}}$$\mathcal{L}$$R$$1.(21)^{\infty}$$1.(2)^{\infty}$$1.(2)^{\infty},$$I_{9},$$I_{5}$$I_{1}$$I_{1}$$I_{3}$$\theta=90^{\mathrm{O}}$

$R$

$C$

$1.2(1)^{\infty}$$1.(21)^{\infty}$$1.2(1)^{\infty},$$I_{10},$$I_{2}$$I_{6}$$I_{1}$$I_{3}$$I_{4}$$\theta=150^{\mathrm{o}}$$R$$1.21(2)^{\infty}$$\mathcal{R}$$1.(21)^{\infty}$$1.21(2)^{\infty},$$I_{11},$$\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}_{\rangle}$$I_{7}$$\theta=30^{\mathrm{O}},$$I_{3}$$90^{\mathrm{o}}$$150^{\mathrm{o}}$$\mathcal{T}$77sequences increase with decrease of from $1.212(1)^{\infty}$ up to. We divide$1.212(1)^{\infty},$$1.(21)^{2k}1\ldots$$1.212(1)n2\ldots$
sequence
symbolinto three groups:, and$n\geq
2,$
$k>0$
with
. We call the regions with these symbol sequences, and, respectively.
$12$ otherThus there are in total twelve regions$i=1,2,$than
Fig.3). The boundary ofandis the triple collision curve with symbol sequence (. .. 1.20). Similarly, the boundaries of and , of and , and ofandare triple collision curves with symbol sequences (. .. 1.210), (... 1.2120),and (. .. 1.21210), respectively. Regions, andwill be called immediate escape regions(called regular regions in Paper I) with symbol sequences, $(. . . 1.2(1)^{\infty})$ , (. . . 1.21 (2) ), and $(. .. 1.212(1)^{\infty})$ , respectively.
...
$R$$1.(21)^{\infty}$$I_{12},$$I_{8}$$I_{4}$$I_{i},$$S(\mathrm{s}\mathrm{e}\mathrm{e}$$\ldots,$$I_{1}$$I_{5}$$I_{2}$$I_{6}$$I_{3}$$I_{7}$$I_{4}$$I_{8}$$I_{9},$$I_{10},$$I_{11}$$I_{12}$$1.(2)^{\infty})$$($

Table V. Subregions and symbol sequences
Name Sequence

next$21)^{lk}22\cdots$$I_{3}\mathrm{o}\mathrm{r}I_{7}$$21)^{2k+1}22\cdots$$I_{1}$$.. \cdot..\cdot\frac{n.\geq 2,k>0,m>0}{I_{1},I1I_{5}^{3}I_{7}1111.21111.\}}$$222222\mathrm{i}^{n_{1}}n_{1,1}(21^{\cdot}2)_{\infty}n_{121}n_{1}nn_{121}22(21(1)1\{^{)}122^{\cdot}m_{2}2m\infty 2^{\cdot}m212.2111.\cdot..\cdot.\cdot$$I_{9}$$I_{11}$1 (2)
1. (2)
1.2 1 (2)

$1\ldots$$I_{10}I_{7}I_{3}I_{2}I_{6}I11$$I_{5}$

escape$I_{9}$

We can further divide $i=1,2,$
into smaller pieces. However, insteadof doing this, we only look into a slightly finer structure of and . This will
be enough for finding periodic symbol sequences which will be done in \S 4.3.
Let us consider . As we described above, symbol sequences inhave$1.(2)^{n}1\ldots$$n\geq2$
the formwithand these considered as decimal numbers decreases when increases along the line
. So the points with symbol$1.(2)^{n}1\ldots$
sequenceswith larger
stays below those with smaller . Let us$n\geq2$
fix
and look into the structure of the point set with symbol sequence$1.(2)^{n}1\ldots$ . The largest symbol sequence (as a decimal number) is.
However, this sequence is unallowable because unallowable word ’22122’ is contained. So the largest allowable symbol sequence is $1.(2)^{n}121(2)^{\infty}$ . Then we have.

$8$

$I_{i},$$\ldots,$$I_{5}$$I_{6}$$I_{5}$$I_{5}$$R$$\theta=30^{\mathrm{O}}$

$n$

$n$

$1.(2)^{n_{1}}(2)^{\infty}$$1.(2)^{n}121(2)^{\infty}>1.(2)^{n_{12}}1(2)^{m_{1}}\ldots>1.(2)^{n_{1}}(21)2m_{22}’\ldots>1.(2)^{n_{1}}(21)\infty>$

where $m\geq 2,$ $m’\geq 1$ , and$k\geq 1$ . In getting the above sequence of inequalities we have made use of Proposition 4.1. For example, sequence $1.(2)^{n}1(21)2m+1\prime 22\ldots$ might have been included$1.(2)^{n}1(21)2k211\ldots>1.(2)^{n}1211\ldots>1.(2)^{n_{12}}(1)^{\infty}$

78$\mathrm{R}$

50

100

theta

150

. Four bands in the left with symbol sequences, and $1.(2)^{9}1\ldots$ , are mapped to four bands inthe right. Corresponding regions have the same color.
Figure 4: Mapping of regions in$1.(2)^{3}1\ldots,$$1.(2)^{5}1\ldots,$$\mathcal{L}$$1.(2)^{7}1\ldots$

in inequalities. However, this is unallowable because it contains unallowable word$,221(21)2m’+122’$ . Now operating
to these sequences, we get 1.21 2,$1.21(2)^{m}1\ldots\in I_{7},1.(21)^{2m_{22}}’\ldots\in I_{3},1.(21)^{2k}211\ldots\in I_{2},1.211\ldots\in I_{6}$, and$1.2(1)^{\infty}\in I_{10}$ . The remaining sequenceis on the stable manifold whichtends to one of the corner of the Schubart region. Now we conclude that orbitsof points whose symbol sequences have the form $1.(2)^{n}1\ldots$ cross in the next in$I6,$ $I10$ or in the boundary
tersection the surface of section either inof and .
Let us confirm the above consideration by numerical integration of orbits.
Let us take points of region whose symbol sequences are $1.(2)^{3}1\ldots,$ $1.(2)^{5}1\ldots$ ,$1.(2)^{7}1\ldots$ , and $1.(2)^{9}1\ldots$ ,and integrate their orbits until the next intersection
with the surface of section. The results are shown in Fig. 4. We see images of theinitial bands have the form also of bands and indeed pass through$I_{7,3,2}II,$,$1.(2)^{n_{1}}(21)\infty$. Now it is clear that
and
, the
, is on the. age of$R$boundary of and .
One can do the same thing for . Here we do not repeat the procedure. One
sees in Table V regions $i=1,2,$ , 12 and the corresponding symbol sequences
together with the regions mapped to in the next intersection. The points in$\sigma^{n}$$($$)^{\infty}\in I_{11}$$1.(21)^{\infty}$$I_{11},$ $I_{7},$ $I_{3},$ $I_{2},$$C$$\mathcal{R}$$\mathcal{L}$$I_{11},$$1.(21)^{\infty}$$I_{10}$$I_{6}$$\mathrm{i}\mathrm{m}$$C$$I_{6}$$I_{i},$$\ldots$$I_{1}$

79
andmove alternately as
and go into . Similarly, the points in, finally make transition
andmove alternately as$I_{2}arrow I_{4}arrow I_{2}arrow I_{4}arrow\ldots$ , finally make transition
and go into .
The initial behavior resembles that of period 2 points near the boundary of the
Schubart region. The motion is almost regular revolving around the Schubart
region and gradually approaching the boundary of ’chaotic regions’ or .
$I_{3}$$I_{1}arrow I_{3}arrow I_{1}arrow I_{3}arrow\ldots$$I_{1}arrow I_{7}$$I_{5}$$I_{2}$$I_{4}$$I_{2}arrow I_{8}$$I_{6}$$I_{5}$$I_{6}$

andare mapped repsectively to
and .
and
are mapped
respectively toand. Finally,
andcorrespond escaping orbits. Thus
we callandthe immediate escape regions.
$I_{7}$$I_{8}$$I_{9},$

4.4$I_{10},$$I_{5}$$I_{9}$$I_{10}$$I_{11}$$I_{12}$$I_{6}$$I_{11}$$I_{12}$$I_{10}$$I_{9}$

Periodic sequences and periodic orbits

In order to obtain periodic sequences, we make in Fig.5 a graph of transitory
relations among from the data in Table V. The necessary condition for a sequence to be periodic is to come back to some starting from there. There are
four shortest round trip paths in the figure:, and. However, the first two should be excluded. As we described in \S 4.3,
these transitions correspond to secular movements in the respective areas, so no
periodic sequences are expected to exist.
Let us consider transition. The form of a sequence in is $1.(2)^{n}1\ldots$ ,$n\geq 2$ , whereas the form is 1.21 2
in . Then, the periodic sequence should, $n\geq 2$ . Similarly, we get a word $12(1)^{n}2$ fromcontain a word 1 2.
We list first several ones in Table VI.
$I_{i}$$I_{i}$$I_{1}-arrow I_{3},$$I_{2}-arrow I_{4},$$I_{5}\neq I_{7}$$I_{6}\neq I_{8}$$I_{5-}arrow I_{7}$

$($

$($

$)^{2}$$\ldots$$I_{5}$$I_{7}$$)^{n_{12}}$$I_{6}\nabla^{-}arrow I_{8}$Table VI. Possible periodic symbol words.

Geometrically, periodic sequences can be obtained as crosspoints of curves orbands. Suppose that a word 1
is a unit of periodic sequence. Periodic sequence should be of the form $(2c_{2^{C_{3}}}\ldots Cn-11)^{\infty}.(2_{Cc}23\cdots Cn-11)\infty$ . Thensequence
is in one of and sequenceisin another . The corresponding periodic sequence is given as crosspoints ofcurves or bands $I(1.(2_{C_{2}c..c_{n}}3\cdot-11)\infty)$ andprovided bothsequences are allowable. Here,means the transformation$(\theta,R)arrow(\pi-\theta,
R)$
of
by
. We show in Fig.5, the cases of
periodic sequences $(22121)^{\infty}.(22121)\infty,$ $(222121)^{\infty}.(222121)\infty$ , and $(2222121)^{\infty}$ .
$(2222121)^{\infty}$ . Crosspoints are seen in smallboxes. $(\theta, R)=(25.20951, 1.8645681)$
is the approximate position of $(22121)^{\infty}.(22121)\infty$ . In the figure, intersecting$c_{2}c_{3}\ldots c_{n}-12$$1.(2c_{2}C_{3\cdots n}C-11)^{\infty}$$I_{i}$$1.(2\overline{C}n-1\cdots\overline{C}3^{\overline{C}}21)^{\infty}$$I_{j}$$I^{t}(1.(2_{\overline{C}_{n-}..\overline{C}}1\cdot 3^{\overline{C}}21)\infty)$$I^{t}(1.(2\overline{C}_{n-}1\cdots\overline{C}3^{\overline{C}}21)\infty)$$I(1.(2_{\overline{C}}n-1\cdots\overline{c}_{3}\overline{C}_{2}1)\infty)$

80

I9$\wedge|$

‘

I ,,

$=$ I

$A$

Figure 5: Graph for transitions among$I_{i}$.

curves are not continuous. This is due to the coarseness of the mesh of orbit
integrations.
There are longer round trip paths. Let us list in Table VII all the possible
round trip paths. Periodic words are obtained in a similar manner. These arealso listed in Table VII. In the table, one seesin the fourth,andfifith, seventh and eighth paths. This means that the corresponding orbits repeat
transitions between these regions as much times as the number of 21 in thesequences.
$I_{3-}arrow I_{1}$$I_{4-}arrow I_{2}$Table VII. Periodic paths and periodic words.
Path

Periodic word$(n, m\geq 2, k, k’>0)$$I_{5}\neq I_{7}$

1 (2) $12$

$I_{6=}arrow I_{8}$$12(1)^{n}2$$I_{5}arrow I_{6}arrow I_{7}arrow I_{5}$$1(2)^{n}12(1)^{m_{22}}$$I_{5}arrow I_{3}\neq I_{1}arrow I_{7}arrow I_{5}$$1(2)^{n_{1(}}21)2k+122$$I_{5}arrow I_{6}arrow I_{3\vee}arrow I_{1}arrow I_{7}arrow I_{5}$$I_{5}arrow I_{2}arrow I_{8}arrow I_{6}arrow I_{7}arrow I_{5}$

1 (2) $2(1)m(21)^{2}k22$
1 (2) I (21) $2(1)^{m_{22}}$$I_{5}arrow I_{2}arrow I_{8}arrow I_{6}arrow I_{3^{\underline{-\Delta}}}I_{1}arrow I_{7}arrow I_{5}$$1(2)^{n_{1}}(21)2k2(1)^{m}(21)2k’22$$I_{6}arrow I_{4}\neq I_{2}arrow I_{8}arrow I_{6}$$12(1)^{n}(21)^{2k}2$812$\mathrm{R}$

1

50

100

theta

150

Figure 6:

Procedure to obtain periodic sequences.
The cases of$(222121)^{\infty}.(222121)\infty$ , and $(2222121)^{\infty}.(2222121)^{\infty}$. The
crosspoints of the curves shown in small boxes correspond to the periodic sequence.
$(22121)^{\infty}.(22121)\infty,$

If future and past periodic sequences occupy bands in the surface of section,then the common points occupy an area in the surface of section. In this case,there may be a stable periodic point in it. On the othere hand, if they are
curves, then intersection is a single point. The point is certainly correspond to an
unstable periodic point. Thus, in any case, we expect that there are an infinite
number of periodic orbits corresponding to periodic symbol sequences in the one
dimensional three-body problem. But we need more precise analysis for a rigorous
result.
Finally, we can get oscillatory sequences. Oscillatory sequences directly specify a group of oscillatory
Tanikawa &Umehara, 1998 for osicllatory
orbits in the planar problem). Consider transitions. The corresponding periodic word is 1 2, $n\geq 2$ . If is fixed for every round trip path,then we get a periodic sequence as before. Let us take a sequence of pos$n_{i}\geq 2$ such that $\lim n_{i}arrow\infty$ ,itive integers
and consider a sequence$1.(2)^{n_{1}}121(2)n_{2}12\ldots 1(2)^{n_{i}}12\ldots$ .
This is an allowable sequence. This is an oscillatory sequence and gives an oscillatory orbit. In a similar manner, we canconstruct oscillatory sequences from other periodic words in Table VII.
$\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{i}\mathrm{t}_{\mathrm{S}}(\mathrm{s}\mathrm{e}\mathrm{e}$$I_{5}-arrow I_{7}$$($$\{n_{i}\}_{i},$$)^{n_{12}}$

$n$

82

4.5

Property of the set of allowable Symbol Sequences

In this subsection, assuming the numerical results, it will be shown that allowable
symbol sequences form a Cantor set. A cylinder is defined as a set of symbol(see Robinson, 1995).
sequences which has a fixed word in a give range of$\mathrm{d}\mathrm{i}\mathrm{g}\dot{\mathrm{i}}\mathrm{t}_{\mathrm{S}}$Assumption 4.2. Unallowable words are all given in Proposition 4.1.

Theorem 4.3. Under Assumption 4.2, the set
and hence in .
forms a Cantor set in$\Omega$of allowable symbol sequences$\Sigma$$\Sigma_{2}$Proof. Cylinders are all open. Therefore,$\Omega=\Sigma_{2}\backslash${$\mathrm{a}\mathrm{l}1$

unallowable cylinders} iscompact.
Take any allowable symbol sequence $s=$ $(. . . s_{-2}1.2s_{12}s, \ldots , s_{n}, \ldots)$ . Let us
as $iarrow\infty$ )suppose that there exists a sequence $1<k_{1}<k_{2}<\ldots$ (such that $s_{k_{i}}=1$ and $s_{k_{i}+1}=2$ . Then is not isolated. For, we have$(. .. 1.2s_{k_{i}2}+s_{k_{i}3}+\cdots)$ and we know that there is an infinite number of allowable
symbol sequences of this form. This implies that there are allowable symbol
sequences in any neighborhood of . is not isoletaed in the case where there
exists $N>0$ such that $s_{n}=1$ or $s_{n}=2$ for $n>N$ . For, we know the existence
for any $n\geq 2$ .
of symbol sequences of the form. (2)In order to show that is totally disconnected, it suffices to show that in any
neighborhood of a sequence in there is a sequence which does not belong to. Using unallowable cylinders with their fixed words in arbitrarily high digits,
we can construct a symbol sequence not belonging to and arbitrarily closea given allowable sequence.
$k_{i}arrow\infty$$\sigma^{k_{i}}s=$

$s$

$s$

$s$

$121\ldots \mathrm{o}\mathrm{r}.(1)^{n}212\ldots$$\Omega$$\Omega$$\Omega$$\Omega$$\mathrm{t}\mathrm{o}\square$

As consequences, we have
Corollary 4.4. There is an uncountable number of non-escape orbits other thanorbits in the Schubart region.

This is obvious because there is an uncountable number of symbol sequences
and a symbol sequence corresponds to more than one
other than (21)mayMoreover,be an uncountable number of curves of initial conthere
orbits.
ditions for non-escape orbits.
$.(21)^{\infty}$

Corollary 4.5. The area of initial positions for a non-repeating symbol sequenceis zero.

This is also obvious because the corresponding orbits appear infinite times indifferent positions of the surface of section.

83

5Conclusions

We have demonstrated that symbolic dynamics is effective in the one-dimensional
three-body problem. Main results are(i) It has been found that there are unallowable sequences of binary collisions,the symplest ones being 2211 and 1122.

(ii) Periodic sequences are found and some of periodic orbits are positioned, thesimplest one being $(22121)^{\infty}.(22121)\infty$ . The existence of oscillatory orbits withcollisions is deduced.
(iii) Qualitative behavior of general orbits on the surface of section is understoodto a certain extent by using symbol sequences and shift operator.

Acknowledgment
One of the authors(K.T.) expresses his thanks to Dr. Shigehiro Ushiki (Kyoto
University) for useful suggestions on symbolic dynamics.

Refererices[1] Block, L.S. and Coppel, W.A., 1992, Dynamics in One Dimension, Lecture Notes in Mathematics Vol. 1513, Springer-Verlag.
[2] Bulirsch, R. and Stoer,, Numeolation Methods. Num. Math., 8, 1-13.
$\mathrm{J}.:(1966)$$7^{\sim}ical$

Treatment of Differential Equations by Extrap-[3] Mikkola, S. and Hietarinta, J., 1989, Celestial Mechanics and Dynamical Astronomy 46,

1-18.

[4] Mikkola, S. and Hietarinta, J., 1990, Celestial Mechanics and Dynamical Astronomy 47,

321-331.

[5] Mikkola, S. and Hietarinta, J., 1991, Celestial Mechanics and Dynamical Astronomy 51,

379-394.

[6] Robinson, C., 1995, Dynamical Systems, CRC Press, Boca Rayton, USA.

[7] Tanikawa, K. and Mikkola, S.,.

$1999(\mathrm{P}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{r}$I), Celestial Mechanics and Dynamical Astron-$omy(\mathrm{S}\mathrm{u}\mathrm{b}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{d})$[8] Tanikawa, K. and Umehara, H., 1998, Celestial Mechanics and Dynamical Astronomy 70,

167-180.

$\sim[9]$Zare, K. and Chesley, S., 1998, Chaos 8, 475-494.