P oceedings o 2nd Ca alan Days on Applied Ma hema ics
c

1995 P esses Uni e si ai es de Pe pignan
ON BIRKHOFF'S CONJECTURE ABOUT CONVEX
BILLIARDS
A. Delshams and R. Ram ez-Ros
Uni e si a Poli ecnica de Ca alunya, Spain
Abs ac .
Bi kho conjec u ed ha he ellip ic billia d
was he only in eg able con ex billia d. He e we p o ea
lo cal e sion o his conjec u e:
any
non- i ial symme ic
en i e p e u ba ion o an ellip ic billia d is non-in eg able.
1. In o duc ion
Le us conside he p oblem o he con ex billia d able": le
C
be
an (analy ic) closed con ex cu e o he plane
R
2
, pa ame e ized by

:
T
;!
C
, whe e
T
=
R
=
2

Z
, in sucha way ha
C
is a eled
coun e clo ckwise. Supp ose ha a ma e ial p oin mo es inside
C
and collides wi h
C
acco ding o he law he angle o incidence
is equal o he angle o eec ion". Following Bi kho 1], his
disc e e dynamical sys em can b e mo deled by an (analy ic) a.p.m.
in an annulus. Mo e p ecisely, conside he annulus
A
=
z
=
(
'
)
2
T

R
:
j
j
<
j
_

(
'
)
jg
, whe e he co o dina e
'
is he
pa ame e on
C
and
=
j
_

(
'
)
j
cos
#
,wi h
#
2
(0

) he angle o
incidence- eec ion o he ma e ial p oin . In his way,weob ain
a map
T
:
A
;!
A
gi en by(
'
)
7;!
(
V
) ha mo dels he
billia d (see Figu e 1).
The unc ion
S
:
(
'
)
2
T
2
:
'
6
=
g;!
R
dened by
S
(
'
) =
j

(
'
)
;

()
j
is a gene a ing unc ion o
T
,i.e.:
@
S
@'
(
'
) =
h

(
'
)
;

()

_

(
'
)
i
j

(
'
)
;

()
j
=
;j
_

(
'
)
j
cos
#
=
;

@
S
@

(
'
) =
h

()
;

(
'
)

_

()
i
j

(
'
)
;

()
j
=
j
_

()
j
cos  =
V
and consequen ly
T
is an a.p.m. (de
DT
(
'
)= 1), and (
'
) a e
canonical conjuga ed co o dina es.
86
A. Delshams and R. Ram ez-Ros

(
'
)
#

()

#

Figu e 1:
T
(
'
)= (
V
), whe e
=
j
_

(
'
)
j
cos
#
and
V
=
j
_

()
j
cos .
Rema k 1.1
The o bi s o
T
(biinni e sequences (
z
n
)
n
2
Z

A
such ha
T
(
z
n
)=
z
n
+1
) a e in one- o-one co esp ondence wi h he
c i ical p oin s o he o mal se ies (called he
ac ion
)
W
((
'
n
)
n
2
Z
)=
X
n
2
Z
S
(
'
n
'
n
+1
)

since hese c i ical p oin s (
'
n
)
n
2
Z

T
sa is y he equa ions
@
1
S
(
'
n
'
n
+1
)+
@
2
S
(
'
n
;
1
'
n
)=0

8
n
2
Z
:
(1)
Thus, ha ing a gene a ing unc ion allows us o wo k wi h only
hal o he co o dina es ( he
base
co o dina es, i.e., he
'
's). The
be
co o dina es (i.e., he
's) a e sup e uous.
Rema k 1.2
Le
C
and
C
0
wo closed con ex cu es such ha
one is he image o he o he by a simila i y. Then he woassoci-
a ed a.p.m.
T
,
T
0
,ha e an equi alen dynamics since he angle o
incidence- eec ion emains unchanged by he simila i y.
The map
T
has no xed p oin s bu is geome ically clea ha
i has p e io dic o bi s o p e io d 2, co esp onding o opp osi e p oin s
wi h he maximum and minimum" dis ance b e ween hem. In
hese o bi s he angle o incidence- eec ion is
=
2and hus
=0.
On Bi kho 's conjec u eabou con ex bil lia ds
87
To s udy he dynamics o hese 2-p e io dic p oin s o
T
,i is
b e e o conside hem as xed p oin s o
T
2
, and s udy
T
2
. Bu
since i is no easy o nd he gene a ing unc ion o
T
2
,we ins ead
in o duce a simplica ion.
Supp ose now ha
C
is
symme ic
wi h ega d o a p oin . By
he p e ious ema k 1.2, we can assume ha his p oin is he o igin.
Then i is p ossible o cho ose a pa ame e iza ion

o
C
such ha

(
'
+

)=
;

(
'
) and he 2-p e io dic o bi s a e o he o m (
'
0

0),
(
'
0
+

0), ha is, wo opp osi e p oin s o e
C
.
Rema k 1.3
We can conside he a iable
'
dened mo dulus

in
he symme ic case.
Le now
R
:
A
;!
A
b e he
in olu ion
R
(
'
)=(
'
+

)
(
R
2
is he iden i y). By he symme y o
C
,
T
and
R
commu e
and i is a commonplace o use his symme y o con e he 2-
pe iodic poin s in o xed p oin s. Conc e ely,we dene a new map
F
:
A
;!
A
by
F
=
R

T
. Since
F
2
=
T
2
, he dynamics o
F
and
T
a e equi alen . Mo eo e , since

( +

)=
;

(), i is easy o
check ha
L
(
'
) =
S
(
'
+

)=
j

(
'
)+

()
j
(2)
is a gene a ing unc ion o
F
, and consequen ly
F
is an a.p.m.
2. Ellip ic Billia ds
The simples examples o con ex cu es a e he ellipses. I is clea
ha he case o a ci cum e ence is e y degene a ed o a billia d,
since i consis s only o 2-p e io dic o bi s. So, le us conside nowa
non-ci cula ellipse:
C
0
=
(
x
2
a
2
+
y
2
b
2
=1
)
=

0
(
'
)=(
a
cos
' b
sin
'
):
'
2
T
g

wi h
a
2
6
=
b
2
. Wi hou loss o gene ali ywe can assume ha
a
2
;
b
2
=1 (wechange he ellipse using a simila i y, i necessa y). Thus
a>
1,
b>
0 and he o ci o he ellipse a e (

1

0). Le us deno e
T
0
:
A
;!
A
he analy ic a.p.m. asso cia ed o he ellipse
C
0
,and
F
0
=
R

T
0
.
88
A. Delshams and R. Ram ez-Ros
I is clea ha he p oin s (0

0) and (

0) o m a 2-p e io dic o bi
o
T
0
ha co esp onds o he e exes (

a
0) o he ellipse, and
hence
z
1
:= (0

0) is a xed p oin o
F
0
.((
=
2

0) is ano he xed
poin o
F
0
.)
Bi kho was he  s one in no icing ha his sys em is
in e-
g able
, i.e., he e exis s an analy ic unc ion
H
(
'
), called
 s
in eg al
, ha p ese es he o bi s o
T
0
:
H

T
0
=
H
. As a conse-
quence, he cu es
H
= cons an
g
a e in a ian unde
T
0
.Ob i-
ously,
H
is also a  s in eg al o
F
0
=
R

T
0
.
In ac , i is no dicul o check ha
H
(
'
) = (sin
2
'
;
2
)
=
2
is such a  s in eg al, unde he assump ion
a
2
;
b
2
= 1. (See, o
ins ance, 2] o he de i a ion o
I
=
;
2
H
+1.)
In addi ion o he in olu ion
R
, he ellipse
C
0
has ano he sym-
me y
R

:
A
;!
A
gi en by
R

(
'
)= (

;
'
).
R

is also an
in olu ion, and mo eo e
F
;
1
0
=
R


F
0

R

,i.e.,
F
0
is
R

- e e sible.
The dynamics o
F
0
, based in he le el cu es o
H
is d awn
in Figu e 2 whe e he esemblance wi h he phase p o ai o a
p endulum shows up clea ly.
0
=
2

;
+
;
;
'
Figu e 2: Phase p o ai o
F
0
;

a e he sepa a ices o
F
0
.
The main p op e ies o
F
0
a e lis ed in he ollowing Lemma,
whose p o o can b e ound in 3].
Lemma 2.1
a)
z
1
=(0

0)
is a sadd le xedpoin o
F
0
and
Spec

DF
0
(
z
1
)] =
 
;
1
g
, wi h

=(
a
+ 1)(
a
;
1)
;
1
>
1
.
On Bi kho 's conjec u eabou con ex bil lia ds
89
Mo eo e , i
h
:= ln

he ol lowing exp essions hold
a
= co h(
h=
2)
 b
= cosech(
h=
2)
:
(3)
b)
;

=
(
'

sin
'
): 0
<'<
g
a e he
sepa a ices
o
F
0
:i
z
2
;

, hen
F
n
0
(
z
)
;!
z
1
when
n
!1
.
c) I


(
)=(
'
(

)


(
))
, whe e
'
(
) = a ccos ( anh
)=
a csin(sech
)
and
(
) = sech
, hen
F
0
(


(
)) =


(
+
h
)
.
In o he wo ds,


a e
na u al pa ame e iza ions
o
;

(wi h
ega d o
F
0
).
d) Le
(
)=
'
(
+
h
)
. Then
b
sin
'
(
) + sin (
)
j

0
(
'
(
)) +

0
((
))
j
=sech(
+
h=
2)
:
(4)
2.1. En i e ellip ic billia d p e u ba ions
Bi kho conjec u ed ha he ellip ic billia d was he only in e-
g able con ex billia d. Ou goal is o see ha his is lo cally ue o
he symme ic billia ds, i.e., any non- i ial symme ic en i e p e -
u ba ion is non-in eg able. (Non- i ial p e u ba ion means no
educible o an ellipse.)
Le
C
"
g
b e an a bi a y amily o p e u ba ions o he ellipse
C
0
, consis ing o analy ic cu es dep ending on a
C
2
wayon
"
and
symme ic wi h ega d o a p oin
O
"
. Le us deno e by
Q

"
he wo
u hes (and opp osi e) p oin s o e
C
"
wi h
Q

0
=(

a
0). Using a
simila i y ha akes
O
"
and
Q

"
o (0

0) and (

a
0) esp ec i ely,
he ini ial amily can b e pu in he ollowing o m
C
0
"
=
(
(
x y
)
2
R
2
:
x
2
a
2
+
y
2
b
2
+
"P
(
x y  "
)= 1
)

(5)
whe e:
I)
P
is analy ic in
x
,
y
and a leas
C
1
in
"
,
II)
P
(
x y  "
)=
P
(
;
x
;
y "
),
III)
P
(
a
0
"
)=
@
y
P
(
a
0
"
)= 0,

90
A. Delshams and R. Ram ez-Ros
o equi alen ly,like
C
0
"
=

(
' "
)=(
a
cos
'
sin
'

b
+
"
(
' "
)]):
'
2
T
g

(6)
whe e:
i)

is analy ic in
'
and a leas
C
1
in
"
,
ii)

is

-p e io dic in
'
.
F om I I I), i ollows ha
P
(
a
cos
' b
sin
' "
)=
p
(
a
cos
' b
sin
' "
)sin
2
'
wi h
p
sa is ying also I), I I). I is easy o check ha , in  s o de
in
"
, he ela ion b e ween

and
P
is gi en by
P
(
a
cos
' b
sin
'
0) =
;
2
b
(
'
0) sin
2
':
(7)
Thus, i
P
(




0) is an en i e unc ion in he a iables
x
and
y
, he
same happ ens o

1
:=

(


0). I is clea ha i

1
=cons an ,
C
0
"
is, in  s o de , a amily o ellipses.
Deni ion 2.1
Le
C
"
g
beape u ba ion o he el lipse
C
0
.We
say ha
C
"
g
is a
non- i ial symme icen i e p e u ba ion
o he
el lipse when i can be pu , using simila i ies, in he o m (6) and
mo eo e ,

1
:=

(


0)
is a non-cons an en i e unc ion.
Le
T
"
b e he map in he annulus asso cia ed o he billia d in
C
"
, whe e he p e u ba ion conside ed is a symme icen i e one
and le
F
"
=
R

T
"
.I
"
is small enough,
C
"
is an analy ic con ex
closed cu e, and hus
F
"
g
j
"
j
1
is a amily o analy ic a.p.m. wi h
gene a ing unc ion
L
"
(
'

"
)=
j

(
' "
)+

(
"
)
j
ha can b e
w i en as
L
"
(
'

"
)=
L
0
(
'
) +
"
L
1
(
'
) +
O
(
"
2
), whe e
L
0
(
'
) =
j

0
(
' "
)+

0
(
"
)
j

L
1
(
'
) =
b
sin
'
+sin
j

0
(
'
)+

0
()
j
sin
'
1
(
'
)+sin

1
()]
:
(8)
Fo
"
small enough, he o igin
z
1
is again a hyp e b olic p oin
o
F
"
. This p oin
z
1
lies in he in e sec ion o he in a ian cu es
W
s
"
,
W
u
"
,such ha any(
z
n
)
n
2
Z
o bi in he mani old
W
s

u
"
ends o
On Bi kho 's conjec u eabou con ex bil lia ds
91
z
1
a an exp onen ial a e as
n
!
+
1

;1
. These in a ian cu es
will no longe coincide o
"
6
= 0. Bu o
"
small enough, he e
exis op en se s
U
s

u
in
W
s

u
"
and analy ic unc ions #
s

u
"

:
I
s

u
!
R
such ha
U
s

u
= G aph (#
s

u
"

), wi h
I
:=
I
s
I
u
o size a li le bi
less han

. Le us see ha he p imi i es

s

u
"

o he unc ions #
s

u
"

ha e a nice in e p e a ion, c ucial o he de i a ion o he Melniko
p o en ial.
P op osi ion 2.1
Le

s

u
"
:
W
s

u
"
!
R
be he analy ic unc ions
denedby

s
"
(
z
s
)=
;
X
n

0
L
"
(
'
s
n
'
s
n
+1
)
 
u
"
(
z
u
)=
X
n<
0
L
"
(
'
u
n
'
u
n
+1
)

(9)
whe e, i
z
s

u
2W
s

u
"
,
z
s

u
n
=(
'
s

u
n

s

u
n
)=
F
n
"
(
z
s

u
)
o
n
2
Z
.Then:
1.

s

u
"
(
z
1
)=0
.
2.

s

u
"

(
'
)=

s

u
"
(
z
s

u
)

8
'
2I
s

u
z
s

u
=(
'
#
s

u
"

(
'
))
.
This esul is a di ec consequence o he a ia ional p inci-
ple (1). Mo eo e , he p op e ies
1.
and
2.
de e mine comple ely
he unc ions

s

u
"
.
To measu e he sepa a ion b e ween in a ian cu es, wewilluse
he
Melniko po en ial
, i.e., a p imi i e o he Melniko unc ion,
ha is no hing else ha he
ac ion
P
L
(
'
n
'
n
+1
"
)a alua ed on
he unp e u b ed sepa a ices ; := ;
+

;
;
up o he  s o de in
"
.
P op osi ion 2.2
Le
L
:;
!
R
be he unc ion denedby
L
(
z
)=
X
n
2
Z
L
1
(
'
n
'
n
+1
)
 z
n
=(
'
n

n
)=
F
n
0
(
z
)
:
(10)
Then:
(i)
L
is wel l-de ned ( he sum in (10) is absolu ely con e gen ),
analy ic and
F
0
-in a ian (i.e.,
L

F
0
=
L
).
(ii) I
L
is no iden ical ly cons an and
"
is smal l enough, he
in a ian cu es
W
u
"
,
W
s
"
in e sec along a ni e con ac
ho-
mo clinic
o bi o
F
"
, and
F
"
is non-in eg able.
92
A. Delshams and R. Ram ez-Ros
P o o o he P op osi ion.
Weonlyske ch he e he p o o . Fo
mo e de ails, see 4].
(i) The sum is absolu ely con e gen since any o bi in he sep-
a a ix ; ends o
z
1
a an exp onen ial a e as
j
n
j!1
and
L
1
(0

0) = 0. Thus
L
is well-dened and analy ic. Mo eo e , a
shi in he index o he sum do es no changes i s alue, so he
F
0
-in a iance o
L
is p o ed.
(ii) Ou measu e o he dis ance b e ween he p e u b ed asymp-
o ic mani olds is he die ence o b e co o dina es ( he
's)
$

(
' "
):=
d

u
"

(
'
)
d
'
;
d

s
"

(
'
)
d
'
=
"M

(
'
)+
O
(
"
2
)
:
M

gi es in  s o de in
"
he dis ance b e ween in a ian cu es,
and hus can b e conside ed as a Melniko unc ion.
Fo a p imi i e


o $

wi h esp ec o
'
,$

:= d


=
d
'
,we
ha e he ollowing exp ession:


(
' "
):=

u
"

(
'
)
;

s
"

(
'
)=
X
n
2
Z
L
"

'

(
n
)
n
'

(
n
)
n
+1


whe e (
'
s

u
n

s

u
n
)=
F
n
"
(
'
d

s

u
"

(
'
)
=
d
'
),

(
n
)=s,i
n

0and

(
n
)=u, i
n<
0.
Le
L

:0

]
!
R
he analy ic unc ion dened by
L

(
'
):=
L
(
'
d

0
(
'
)
=
d
'
), whe e

0
(
'
)=
;
cos
'
. Using he uni o m ap-
p oxima ions o
W
s

u
"
by;and he a ia ional p inciple, i u ns
ou ha


(
' "
)=
"L

(
'
)+
O
(
"
2
)
:
Thus,
M

=d
L

=
d
'
,and now i is clea ha i
L
is no iden i-
cally cons an , he same will happ en o
L

. Finally, one can see ha
M

is no ze o, analy ic, and has a ze o o ni e o de , using he ac
ha
L


'
is
h
-p e io dic, whe e

(
)=(
'
(
)

(
)). Consequen ly,
o
"
small enough, he in a ian cu es
W
u
"
,
W
s
"
in e sec along a
ni e con ac
homoclinic
o bi o
F
"
. By a esul o Cushman 5],
F
"
is non-in eg able.
2
F om nowon,we conside only ;
+
. Using he pa ame e iza ion
p o ided by (2.1), we can w i e ;
+
=

+
(
)=(
'
(
)

(
))
g
,and
conside he Melniko po en ial
L
(
)=
X
n
2
Z
(
+
hn
)

wi h
(
)=
L
1
(
'
(
)
'
(
+
h
))

On Bi kho 's conjec u eabou con ex bil lia ds
93
whe e o simplici y o no a ion, wew i e
L
ins ead o
L



,wi h
L
1
as gi en in equa ion (8). Using o mula (4), we a i ea an
equi alen exp ession o
:
(
) = sech(
+
h
2
)sech(
)

1
(
'
(
)) + sech(
+
h
)

1
(
'
(
+
h
))]
:
(11)
2.2. Non-in eg able billia ds
The aim o his pap e is o p o e he ollowing esul .
Theo em 2.1
Le
C
"
g
be any non- i ial symme ic en i epe -
u ba ion o an el lipse. Then he bil lia din
C
"
is non-in eg able o
0
<
j
"
j
1
.
P o o o he Theo em.
By p op osi ion 2.2, we only ha e o
p o e ha
L
is non cons an . To hisend,we will use he ollowing
esul .
Lemma 2.2
Inacomplex neighbou hood
D
o
=

i
=
2
, he eex-
is s a analy ic unc ion
G
such ha
L
(
)
eads as
L
(
)=
2
a
b
sech(
+
h=
2) sech(
;
h=
2)

1
(
'
(
)) +
G
(
)
:
Accep ing o he momen his lemma, and using he ac ha
sech(
+
h=
2) sech(
;
h=
2) is analy ic non ze o on
=

i
=
2, i
u ns ou ha
=

i
=
2 is a singula p oin o
L
(
) i and only
i he same happ ens o

1
(
'
(
)). Bu byhyp o hesis,

1
is a

-
p e io dic en i e unc ion, and by lemma 2.1, sin
'
(
) = sech(
)and
cos
'
(
) = anh (
)ha e simple p oles a
=

i
=
2 and no mo e
singula i ies on
=
=
=
2. Since

1
is non cons an ,
=

i
=
2isa
singula p oin o

1
(
'
(
)) and consequen ly
L
(
) is no a cons an .
2
P o o o Lemma2.2.
Lo oking a he exp ession (11) o
,and
using he ac ha

1
is en i e, he only p ossible singula i ies o
wi h
=
=
=
2 a e
=

i
=
2,
=

i
=
2
;
h
,and
=

i
=
2
;
h=
2.
Ne e heless,
=

i
=
2
;
h=
2isa emo able singula i y, since i is
a simple p ole o sech(
+
h=
2), and also a ze o o sech(
)

1
(
'
(
)) +
sech(
+
h
)

1
(
'
(
+
h
)). So we can w i e
L
(
)=
X
n
2
Z
(
+
nh
)=
(
;
h
)+
(
)+
G
1
(
)