Physics The Baker Transformation and a Mapping Associated to the

Communications

in

Commun. Math.

Phys.

80,

465-476 (1981)

MSthβΓΠStίCQl

Physics

©

Springer-Verlag

1981

The

Baker

Transformation

and a

Mapping

Associated

to the

Restricted

Three

Body

Problem

Robert

L. Devaney

1

*

Department

of Mathematics, University of Maryland, College Park, MD 20742, USA

Abstract. The nonlinear mapping of the plane

was recently introduced by

Henon

as an asymptotic form of the equations

of motion of the restricted three body problem. This

is

an area preserving

diffeomorphism, except along the

x-axis

where the mapping is singular. We

show that this mapping exhibits

a

type

of

stochastic behavior known

as

topological transitivity, by showing that

it is

topologically conjugate to the

well

known baker transformation. Consequently, periodic points are dense

in

the plane and there

is

also

a

dense orbit. We note that the baker

transformation

also preserves Lebesgue measure and

is

ergodic,

so

this

raises interesting open questions about the ergodic properties of the non-

linear mapping.

Consider the nonlinear mapping of the plane (x

1?

y

1

) = F(x

0

, y

0

)

given

by

1

yi=y

o

-χo----

This mapping was recently encountered by

Henon

[2] in his studies

of

the

restricted three body problem

of

classical mechanics. Roughly speaking, this

mapping

is

an asymptotic form of the equations of motion which

is

related to

1

Current address: Department of Mathematics, Boston University, Boston, MA 02215, USA

*

Partially supported by NSF

Grant

MCS

79-00430

0010-3616/81/0080/0465/S02.40

466 R.L. Devaney

the

ultimate behavior of families of periodic solutions as the mass-ratio tends to

zero.

Henon's

numerical work indicates that this mapping has highly unstable

and

chaotic behavior, and he has conjectured that it is equivalent to the well-

known baker transformation. Our goal in this paper is to

verify

this conjecture

by constructing a topological conjugacy between F and the baker transfor-

mation.

The

conjugacy is given in fact by a natural mapping: to any point peR

2

,

we associate a doubly infinite sequence s(p) of +Γs and — Γs according to

whether

successive iterates of p lie above or below the

x-axis.

The mapping is

not

defined when

j/

= 0, so certain orbits terminate when they meet this line.

We can extend the conjugacy to these points by assigning a terminating

sequence of + Γs and — Γs to them.

There

is a natural identification of these sequences with the open square

\u\,

\v\<l in the plane.

Under

this identification, the mapping F goes over to

the

baker transformation. Thus the coding s

gives

a complete description of the

associated dynamical system, since we may take the baker transformation to be

completely understood.

Theorem

A. The

mapping

F of Hέnon is

topologically

conjugate

to the baker

transformation

of the

open

square

0^\u\,

\v\

< 1 in the plane.

The

dynamics of the baker transformation thus yield immediately the

following corollaries.

Corollary

B.

Periodic

points

for F are

dense

in the plane. There are exactly 2

n

—

2

fixed

points

for F

n

for each n^l.

Moreover,

each

periodic

point

is

hyper-

bolic.

Corollary

C. There is a

point

peR

2

whose

F-orbit

is

dense

in R

2

. Equivalently,

F is

topologically

transitive.

The

x-axis

is a singular set for F since orbits which meet this set cannot be

continued.

Although the set of all such singular orbits is of measure zero in the

plane,

it nevertheless is dense.

Corollary

D. Let S

denote

the set of

points

in R

2

whose

orbits

terminate at the

singular

set

j/

= 0. Then S is

dense

in R

2

.

The

proofs of each of these corollaries follow immediately from the exis-

tence

of the topological conjugacy given by Theorem A and the corresponding

statements

for the baker transformation.

The

mapping F preserves Lebesgue measure in the plane. This raises

several intriguing questions about the ergodic properties of this mapping. Is F

ergodic? It is well-known that the baker transformation preserves Lebesgue

measure in the square and that it is ergodic. So our conjugacy

gives

a different

(infinite)

measure on the square which arises naturally in the context of

mechanics.

It would be interesting to know exactly what this measure on the

square is.

Baker

Transformation 467

1.

The

Baker

Transformation

F

is a

real analytic, area preserving mapping defined

on R

2

—

(y

= 0).

Its

inverse

is also real analytic

and is

given

by (x_

x

, y_

ί

)

= F~

1

(x

0

,

y

0

)

where

1

X 1 ΛΛ

Hence

F~

x

is not

defined

on the

line

y

o

=

—

x

0

.

We

will

find

it

convenient

to

denote

the

point F

j

(x

0

,

y

0

) by (xj,

j^ )

for jeΈ.

Let peR

2

.

If F

k

(p) is

defined

for all keZ, we may

assign

a

doubly infinite

sequence

s(p) of + Γs and

—

Γs to p via the

rule

φ)

=

(...s_

2

,s_

1

,s

o

;s

1

,5

2

,

...),

where

Some F-orbits terminate when j^(p) =

O

under forward iteration,

or

when

^(p)

=

— Xy(p) under backward iteration.

To

these orbits

we

assign

a

terminating

sequence

of

the form

if

j;_

fc

(p)=

—x_

k

(p\

or a

sequence

of

the form

[O,^.,

...,5_

1

s

0

;5

1

,5

2

, ...),

if 3>j+i(p) =

0

with )^1.

We

also

allow

finite sequences

of

the form

[0,

1

s_

j

,

...,s_

l9

s

o

ιs

u

...,s

k

,0].

So

for

each peR

2

, there

is

assigned

a

sequence

fo the

form

[0,s_

7

.,

...,s

o

;s

l9

...,5

k

,0]

9

where; and/or

fe may be

infinite.

We

assign

the

sequence

[0; 0] to the

origin.

Let

Σ

denote

the set of all

possible sequences

of the

above type, excluding

those which terminate

(to the

right and/or left) with

an

infinite string

of + Γs

or

—

Γs.

That

is, Σ

consists

of all

sequences except those

of

the form

(...,α,α,α,5_

j?

...

9

s

0

;s

ί9

...,s

k

,0]

[O

9

s_

j9

...

9

s_

ί9

s

0

;s

l9

...,s

k

,

α, α, α, ...)

with

an

infinite string

of α's. Σ may be

mapped

onto

the

open square

0^|w|,

\v\<ί

in the

plane

via the

transformation

k

s ~

k

s

/=

1

l

i= 0

468 R.L. Devaney

Note

that

sequences

of the

form

[0; s

u

..., s

ki

0] are

mapped

to v = 0,

while

sequences

of the

form

[0,s_

j9

...,s

o

;0] are

mapped

to

w

= 0. The

boundaries

|M|

= 1 and |u| = l are

excluded since sequences

of the

forms

(...,α,α,α;s

o

,...,s

k

,0]

[0,

s_j,

...,s

o

;α,α,α,

...)

are

excluded.

Also,

one

should think

of the

disallowed sequences

of the

forms

[0,

s_

j9

...,s_

1

,s

0

;

...,s

k

, +1,

-1, -1,

-1,...)

and

[0,

s_

j?

...,s_

1}

s

0

;

...,s

k

,

-1,

+1, +1,

+1,...)

as being identified with

the

allowed sequence

this

being the usual identification which

yields

the dyadic rationals. There is a

similar identification for negative indices.

On

Σ we may define the usual shift automorphism by

σ([0,s_

j9

...,s_

l9

s

o

;s

ί9

...,s

k

,0])

=

[0,s_

j

, ...

9

s_

ί

;s

θ9

s

l9

...,s

k

,0]

provided s

o

φ0. On the square, σ is represented by the well-known baker

transformation

(see, for example, [4, p. 63] or [1]). If we impose the topology

of the square on Σ, then σ is a homeomorphism as long as s

0

+ 0, where σ is

undefined.

The range of σ includes all sequences except those with s

ί

=0.

In

the next sections we

will

prove that s

gives

a topological conjugacy

between F and σ, i.e., s is a homeomorphism which makes the

following

diagram commute

2.

Hyperbolicity

In

this section we prove that there

exists

at least one point in the plane which

corresponds

to any allowable sequence. Our main tool is the (non-uniform)

hyperbolicity of F.

Let

(ξ

0

, η

Q

) be a tangent vector to IR

2

. We denote the forward image

dF(ξ

o

,ηo)

by (ξ^ηj and the

inverse image

dF-

1

{ξ

0

,η

Q

) by

{ξ_

ί

,η_

1

).

In

each

tangent

space, define the sectors

Baker

Transformation

469

and let

(χ,y)

X>y

(χ,y)

So

the

unstable

sectors

S

u

consist

of the

second

and

fourth

quadrants

in

each

tangent

space,

while S

s

consists

of the

other

quadrants.

Let

||(^o?^o)ll

denote

the

sup

norm

on

each

tangent

space,

i.e.,

\\(ξ

0

,

η

o

)\\

=

sup(|£

0

|

The

proof

of the following

proposition

is

straightforward.

Proposition 1.

ii) For(ξ

0

,η

0

)eS»

(Xiy)

,

\\dF(ξ

0

, η

o

)\\^mi

iii) For (ξ

0

,

η

o

)eS

s

(x,

y),

Remark.

This proposition implies that F is non-uniformly hyperbolic; the rate

of expansion in the unstable sector tends to 1 as

|>Ί->oo.

The existence of

invariant manifolds almost

everywhere

for such mappings is a

difficult

problem

studied by Pesin [6].

Definition.

A smooth curve y(t) is called an

unstable

curve

if γ(ή

lies

in the

interior of S"

(ί)

for all t. An unstable curve of the form (t,f(f)) for

—

oo<ί<oo

is called an

unstable

separatrix

if lim f(ή=+ao. Similarly, y(t) is a

stable

ί-> ± 00

curve

if y'(t) lies in the

interior

of S

s

γ{t)

for all t. And a stable curve of the

form

(t,f(ή) is a

stable

separatrix

if either

i) lim/(ί) = oo and lim /(ί) = 0, or

t—>

00 t—> — 00

ii) lim/(ί) = 0 and lim f(t)=—00.

ί-> 00 t-> — 00

See

Fig. 1.

Proposition 2. // y(t) is an unstable (respectively stable) curve, then so is

F(y(t))

(respectively,

F~

1

(y(t))).

Moreover, if y'(t) is contained in a particular quadrant

of S

u

y{t)

(respectively,

S

s

y{t)

),

then

dF(y

f

(ή)

(respectively,

dF~

1

(y

f

(t)))

is contained in

the

same quadrant.

Proof.

One

checks

easily

that

ξ

x

and ξ_

1

have the

same

sign as ξ

0

, and

that

r\

γ

or

η_

ί

have the

same

sign as η

0

,

provided

(ξ

o

,η

o

)

lies in the

appropriate

sector.

Proposition 3. i) Let y(t) be an unstable separatίx.

Then

F(y(ή)

is a pair of

unstable

separatrices, one on each side of the line y =

—

x.

ii)

Let μ(t) be a stable separatrix.

Then

F~

1

(μ(t))

is a pair of stable separa-

trices, one of each of the above types fi.e., one on each side of the

x-axis).

Proof

We prove i);

case

ii) is similar. Let y(ή = (t,f(t)) for ίeR and suppose ί*

is the unique point for which /(ί*) = 0. Let y

1

(ή = (t,f(ή) with ί<ί* and γ

2

(t)

470

R.L.

Devaney

Fig.

1. y is an unstable separatrix, while μ

x

and μ

2

are stable separatrices

=

(

ί

J

/(0) with

t>t*'

Then

proposition.

and

F(y

2

(ί))

satisfy

the conclusion of the

Definition. An

unstable

strip

is a closed region in the plane bounded by a pair

of non-intersecting unstable separatrices, or else the closed region to the left or

right of a single unstable separatrix. Stable

strips

are defined analogously.

Corollary

4. Let V be an unstable

strip.

Then F(V) is a pair of unstable

strips,

one on each side of y =

—

x. If H is a stable

strip,

then F~

1

(H) is a pair of stable

strips,

one on each side of y = 0.

Note

that stable and unstable separatrices must meet at a unique point.

Consequently,

the intersection of stable and unstable strips is non-empty and

bounded

by stable and unstable curves.

We now prove that there exists at least one point in 1R

2

corresponding to

any

sequence of the form

(...s_

2

,s_

ί

,s

0

;

s

l5

s

2

,...)

as long as ^φSy for some

pair

i,j^.O

and s

k

+

s^

for another pair fc, /<0. The proof is analogous to the

proof of the conjugacy of the well-known Smale "horseshoe" diffeomorphism

with the binary shift, so we merely sketch the proof. Further details can be

found

in Smale's original paper [7], or in the books of Moser [4] or Nitecki

[5].

Let

Q

1

(respectively, Q_

1

) denote the upper (respectively, lower) half plane.

The

closures of Q

x

and Q_

ί

are stable strips, while the closures of F(Q

1

) and

F(Q_

1

) are unstable strips bounded by the line y=

—

x.

Define

inductively

=

closure(Q

So

nF-

1

(Hs-

1

...

s

-

n

))

Baker

Transformation

471

One

checks immediately that the

H

SQS

_

ί

^

s

_

n

form a nested sequence of stable

strips. See, for example, Moser [4, p. 172]. Moreover, any point on the bound-

ary of

H

SoS

χ s n

is mapped to y = 0 by some iterate of F.

Similarly, define

^

1

...

Sn

=

closure(F(β

Sl

)n...nF"(β

an

))

=

closure(F(β

Sl

)nF(7

S2

..J).

These are nested unstable strips whose boundaries consist of unstable separa-

trices mapped to y=

—

x by some iterate of F"

1

.

Lemma

5. //

some

S φSy,

then

V

Si Sn

with

n^ί, j^l is

bounded

by two

unstable

separatrίces.

Proof.

For

fixed

n, the set of unstable strips

V

So

^

Sn

fills

the plane, and any two

strips meet only along at most one unstable separatrix in their boundaries. One

checks

easily

that

F

So

_

Sn

is bounded by only one unstable separatrix when Sj=l

or

Sj

= — 1 for all j, 1 ^j ^ n. Hence all other strips lie between these two, and

hence

they are bounded by two unstable separatrices.

Now

suppose (s) =

(...s_

2

,

s_

ί9

s

o

; s

1?

s

2

...)

is any sequence

satisfying

s

t

+

Sj

for some ί, j > 0 and s

k

φ s

£

for some k, £ ^ 0. Consider the sets

V

Sι

...

Sn

πjF/

SoS

_

ι

s

_

n

. By the lemma, these form a nested sequence of compact sets.

The

intersection

Π

(^,.....nH

Soϊ

.,..,.„)

n=

1

then

contains at least one point. This point must then have associated se-

quence

(5), as the k

ίh

iterate of this point

lies

in β

s

_

k

. This proves the existence

of a point corresponding to each doubly infinite sequence.

For

the terminating sequences, we argue as

follows.

To find a point corre-

sponding to the one sided terminating sequence

(...s_

2

,s_

l5

s

0

;

s

l9

...,s

n

,0]

we

first

note that

V

Si

^

Sn

is bounded by one or two unstable separatrices. Only

one

of these curves is mapped to the line y=

—

x by F

1

"". Call this curve y.

Then

any point in

(.QΛ —-

n=0

is associated to the sequence

(...s_

2

,s_

1

,s

o

;s

1

,s

2

,

...,s

Λ

,0],

The

other types of terminating sequences are handled similarly.

We conclude this section by proving that s

gives

a continuous mapping

from R

2

to Σ. If s(p) is a non-terminating sequence, then our construction of

the

stable and unstable strips

shows

that s is continuous at p. So we confine

our

attention to terminating sequences.

First

suppose p*=(x*,0) is on the

x-axis.

We

will

prove that s is con-

tinuous

at p*; all other one- or two-sided terminating sequences are handled

similarly.

472 R.L. Devaney

Lemma

6. Let M> 1.

Suppose

p = (x

0

, y

0

). Then

there

exists

ε>0

such

that

i) 7/0<j;

0

<ε

and \x*—

x

o

|<cβ, £/ιen x

x

>M

and y

1

< —M.

ii)

7/

— ε<y

o

<0

and \x*

— x

o

\<ε, then

x

< —M and

y

ί

>M.

Proof.

For part i) we choose ε so that x*H

2ε>M.

Then we have

o

x

ί

=x

0

-\

>xg —εH—>M + ε>M

Part

ii)

follows

similarly.

Remark. This lemma

shows

that points near the

x-axis

are mapped arbitrarily

far

away

by F, with y

ί

approaching either + oo or — oo, depending upon the

sign

ofjv

Now suppose s(p*) = [0; sf, s*> ...) and keZ

+

.

We claim that there is a

neighborhood W of /?* such that if p = (x

o

,y

o

)eW and y

o

ή=0, then s(p)

=

(...s_

1

,s

0

;

s

ί

...) satisfies

i)

Sj

= sf for

l^j^fc

ii) if y

o

>0, then s

o

= +1 while s_

t

= -1 for

l^i^fe.

iii) if y

o

<0, then 5

0

= — 1 while s_

t

= +1 for

l^irgfc.

In

the topology of the square, this means that s(p) is close to s(p*), proving

continuity.

Choose M>k, so that 2M

—

k>0. If _y

o

>0, then by the lemma, there

exists

ε>0 such that if \x*— x

o

\<ε and y

o

<ε, then x

1

>M and

y

1

<—M.

We also

have

x

2

= x

λ

H

>M

)Ί

M

Continuing

inductively,

it

follows

that

for

2^/^/c,

we

have

x

i+1

>M-ϊ>0

y

ί+1

<-2M

+ i<0.

Hence

s_

t

(p)= —1 for l^i^k and s

o

(p)= +1 as required.

If y

0

< 0, the argument is similar.

3.

Proof

of

Theorem

A

In

this section we complete the proof of Theorem A by showing that s is 1 — 1.

Suppose p = (xo,yo)- We show

first

that the orbit of p either changes

sign

infinitely often or

else

meets y = 0. This

follows

from

several

lemmas.

Lemma

1.

Suppose

x

o

^0,

y

o

>O.

Then

there

exists

n>0

such

that

y

n

^0. //

x

o

^0,

y

o

<0,

there

exists

n'>0

such

that

JV^O.

Baker

Transformation 473

Proof

We prove the

first

statement; the proof of the second is similar. Suppose

y

n

>0

for all n. Then we have

1

n n-1 ^^ n-ί '" 0 =

We also have

and

by induction

y

n

<y

0

-nx

0

.

Therefore,

if x

o

>O, we have a contradiction. If x

o

= 0, then x

1

>0 and we may

apply the above argument using x

1

instead, again providing a contradiction.

Lemma

2.

Suppose

x

o

<0 and y

o

>0. Then

there

exists

n>0

such

that

either

y

n

^0

or

else

x

n

^0. If x

o

>0 and y

o

<0, then

there

exists

n'>0

such

that

either

y

n

,Ξ^O

or

else

x

n

,^0.

Proof

Again we prove only the

first

statement. Assume y

n

>0 for all n. As in

Lemma

1 we have x

n

>x

0

and

0<y

n

<y

o

<nx

o

.

Now

1

11

%9

— ^

1

i -^ Xπ "ι I

yi

yo y

o

-

χ

o

and

by induction,

1

1 1

+ +++

Now

the series ^ diverges, so that x

M

->oo, which

gives

a con-

tradictioa

-o>Ό-^o

Combining Lemmas 1 and 2, we find that the orbit of p must continually

cross the

x-axis,

or else meet it after a finite number of iterates. It is for this

reason that sequences which end with an infinite sequence of + Γs or

—

Γs are

disallowed.

For

the remainder of this section we let p = (χ

09

y

0

) and p' = (x'

o

, y'

o

). Define

Δx

n

= x'

n

—

x

n

and Δy

n

= y'

n

—

y

n

for each

neΊL.

Lemma

3.

Suppose

s(p) = s(p

r

).

i) // zJx

o

<0 and zly

o

>0, then Λx

n

<Λx

n

_

1

and

Ay

n

>Ay

n

_

1

for all n>0.

Furthermore,

lim Ay

n

=co.

ii) // zJx

o

>0, zlj;

o

>0, then Δx__

n

>Δx

0

and Λy_

n

>Δy_

n+1

for all n. Fur-

thermore,

lim Δy_

n

= oo.

W->

00

Remark. In the

first

case we are assuming that the straight line connecting p to

p'

is an unstable curve, while in the second case, it is a stable curve. If p and p'

lie on the same horizontal or vertical line, the lemma also

follows

by applying

it to F(p) and F(p

f

) or F"

1

^) and F~ V)

474

R.L.

Devaney

Proof.

Again

we

prove only part

i).

Since

p and p'

have

the

same sequences,

it

follows that

y'

n

and

y

n

have

the

same signs. Hence

Ax

1

=Δx

0

-{—

<zlx

o

<0

Δy

1

=

Δy

o

-Δx

1

>Δy

o

-Δx

o

>Δy

o

.

By induction,

if Δy

n

_

x

>0,

then

J

L<o

y'n-i y«-i

so

that

Δx

n

<Δx

n

_

1

<Δx

0

.

Hence

Δy

n

=

Δy

n

_

ί

-Δx

n

>Δy

Ό

-(n-l)Ax

0

-Ax

0

.

It

follows that

Ay

n

->oo.

Remark.

It

follows from Lemma

3

that

if

s(p) = s(p')

and the

straight line

joining

p to p' is an

unstable (respectively, stable) curve,

then

so is the

straight

line

joining

F

n

(p) to

F

n

(p')

for all n>0

(respectively,

n<0).

Lemma

4.

Suppose

p and p'

have

the

same

two-sided

terminating sequence. Then

P =

P'

Proof.

The

result

is

clear

if p

= 0.

lϊ y

k

= y'

k

=

O

for

some fc>0

5

then

the

straight

line

joining F

h

~

1

(p)

to

F

k

~

1

(p

r

)

is a

stable curve since

y

k

_

1

and

y'

]ί

_

1

have

the

same sign.

But

then

the

remark above implies that

the

straight line joining

F

k

~

a

(p)

to

F

k

~

a

(p')

for all α^

1

is a

stable curve. Hence this line

can

never have

slope —1,

and so

there does

not

exist

α>0

such that

y

k

_

a

=

—x

k

_

a

and

y'

k

_

Λ

=

—

x'-k-a contradicting

the

assumption that

s(p) is a

two-sided terminating

sequence.

Lemma

5.

Suppose s(p) = s(p')

and

Δx

o

<0, Δy

o

>0. Then

for any M>0

there

exists

j,

fe^O

such that y'j>M

and

y_

k

< —M.

Proof

Assume

y\<M for all i^O. By

Lemma

3, Δy

n

increases

to oo as

n^oo.

Hence

there exists

iV§:0

such that

for n^N,

Δy

n

>M.

By

Lemmas

1 and 2,

there

exists fc^JV such that y'

k

>0.

But

then

y

k

= y'

k

—

Δy

k

<0, contradicting

the

fact that s(p)

=

s(p

f

).

Lemma

6.

Suppose s(p) = s(p')

and

Δx

o

<0, Δy

o

>0. Then

there

exists

fc>0 such

that

ii) x

k

,x

k+l

>0.

iϋ)

3>k>3>k+i>°

Proof.

By

Lemma

5,

there exists

j > 0

such that

Baker Transformation

and

x'j<0. Now y

f

j+1

>y

f

j iϊ x'

j+ί

<0, so there

exists

α^O such that x'

}

_

x;

+α+1

>0.Then

475

<0

but

,

Ax

Hence

x'

j+a

>

Ax

.

Also

x'

j+a

_

1

>

—

fAx. Let k=j +

oc

—

1. Then we have

i

< ^1

+1

^ 0 and y'

k +1

> y^ > 0. Furthermore,

3

This completes the

proof.

Lemma

7.

Suppose

s(p) = s(p') and Ax

0

,

Ay

o

>0.

Then

there

exists

k<0

such

that

iii)

y^Λ.^O.

Proo/.

The

proof

is

similar

to the

preceding

lemma

and

hence

is

omitted.

Proof

of

uniqueness.

Suppose s(p) = s(p') and

Ax

o

<0,

Ay

0

>0.

The other case is

handled

similarly, so the proof is omitted.

Choose fc^O as in Lemma 6, and let ζ be the straight line connecting F

k

(p)

to

F

k+1

(p'). See Fig. 2. We assume that the tangent vectors ζ' lie in the second

quadrant,

so that dF

a

(ζ

f

) also lies in the second quadrant for α>0. In the

terminology of Sect. 1, ζ is an unstable curve. By Lemma 1, there

exists

n>0

such that y

n

(F

k+ί

(p))<0 but y

n

(F

k

(p)) = y

n

-ΛF

k+1

(p))>0. Hence y

n

(F

k+1

(p'))<0.

Then

F

n

(ζ) is an unstable curve connecting F

n+k

(p) to F

n+k+1

(p'). Hence

tangent

vectors to this curve cannot lie in the second quadrant. This contradic-

tion

establishes the result.

Fig. 2

476 R.L. Devaney

Acknowledgements. It is my pleasure to thank J. Curry and M. Henon for pointing out the

numerical studies of this mapping to me, M. Brin and A. Katok for many stimulating discussions

while this paper was written, and F. Przytycki for his helpful comments on the final manuscript.

References

1.

Arnold, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. New York: Benjamin 1968

2.

Henon, M.: Notes on the restricted three body problem. Mimeographed

3.

Katok, A., Strelcyn, J.-M.: Invariant manifolds for smooth maps with singularities. (To appear)

4.

Moser, J.: Stable and random motions in dynamical systems. Annals of Mathematics Studies,

No.

77. Princeton, New Jersey: Princeton University Press 1973

5.

Nitecki, Z.: Differentiable Dynamics. Cambridge, Massachusetts: MIT Press 1971

6. Pesin,

Ja.B.:

Math. USSR Izvest. 10, 1261-1305 (1976)

7.

Smale, S.: Diffeomorphisms with many periodic points. In: Differential and Combinatorial

Topology, 63-80. Princeton, New Jersey: Princeton University Press 1965

Communicated by D. Ruelle

Received December 4, 1980