Publicacions
Ma emá iques,
Vol
37
(1993),
95-109
.
AN
INDESTRUCTIBLE
BLASCHKE
PRODUCT
IN
THE
LITTLE
BLOCH
SPACE
A
bs ac
CHRISTOPHER
J
.
BISHOP
The
li le
Bloch
space,
13
0
,
is
he
space
o
all
holomo phic
unc-
ions
on
he
uni
disk
such
ha
lim1
z
1-
l
( '(z)j(1
-
Iz12)
=
0
.
Fini e
Blaschke
p oduc s
a e
clea ly
in
130,
bu
examples
o
in-
ini e
p oduc s
in
80
a e
mo e
di icul
o
ob ain
( he e
a e
now
se e al
cons uc ions
due
o
Sa ason,
S ephenson
and
he
au ho ,
among
o he s)
.
S ephenson
has
asked
whe he
130
con ains
an
in ini e,
indes uc ible
Blaschke
p oduc ,
Le
.,
a
Blaschke p od-
uc
B
so
ha
(B(z)
-
a)/(1
-
QB(z)),
is
also
a
Blaschke
p od-
uc
o
e e y
a E
D
.
In
his
pape
we
gi e
an
a i ma i e
an-
swe o
his
ques ion
by
cons uc ing
such
a
Blaschke
p oduc
.
We
also
answe
a
ques ion
o
Ca mona
and
Cu í by
cons uc -
ing
a
VMO
unc ion,
,
so
ha
Il
J¡
.
=
1
and
whose
ange
se ,
R( ,
a)
=
{w
:
he e
exis s
z
n
,
~
a,
(z~)
=
w},
equals
he
open
uni
disk
o
e e y
a
E
T
.
1
.
In oduc ion
Le
D
=
iz1
<
1}
deno e
he
uni disk
.
The
li le
Bloch
space,
B
O
,
is
he
space
o
holomo phic
unc ions
on
D
such
ha
lim

1
'(z)1(1
-
Iz12)
=
0
.
Iz1-1
Basic
ac s
abou
B
O
can
be
ound
in
[21
.
A
Blaschke
p oduc
is
a
holo-
mo phic
unc ion
o
he
o m
This
wo k was
suppo ed
by
NSF
G an
DMS
91-00671
96

C
.
J
.
BISHOP
B(z)
=
11
~
z
n
-
z
Iznl
n
1-znz
zn
,
whe e
E(1
-
Izn1)
G
oo
.
Fini e
Blaschke p oduc s
a e
clea ly
in
B
O
,
bu
examples
o
in ini e
p oduc s
in
BO
a e
no
so
ob ious
.
Such
unc ions
do
exis ,
as
is
shown
in
[1], [8],
[10]
.
A
mo e
explici
example,
as well as
a
cha ac e iza ion
o
such p oduc s
in
e ms
o
he
ze o
sequence
{z
n
},
has
been
gi en
in
[3]
.
Tha
esul
answe s
se e al
ques ions
abou
130,
bu
does
no
esol e
he
ollowing
ques ion
om
[10]
:
does
B
o
con ain
an
in ini e,
indes uc ible
Blaschke
p oduc ,
Le
.,
a
Blaschke
p oduc
B
so ha
_
B(z)
-
a
Ba(z)

1
-
úB(z)'
is
also
a
Blaschke
p oduc
o
e e y
a
ED?
The
ques ion
a ises
because
o
F os man's
heo em
.
An
inne
unc ion
F
is
a
holomo phic
unc ion
on
D
wi h
bounda y
alues
o
absolu e
alue
1
a
.e
.
on
T
.
Any
such
unc ion
can
be
w i en
as
a
p oduc
_
II

¡o
F(z)
=
B(z)S(z)
=
( l
1
n
zn
z
~)(exp(
-
e
ie
±
zdp(e)),
o
a
Blaschke
p oduc
and
a
singula
inne
unc ion
(M
is
a
ini e,
posi i e
measu e
singula
o
dB)
.
F os man's
heo em
s a es
ha
o
any
inne
unc ion
F
on
D,
Fa

F(z)
-
a
(z)
=
1
-
úF(z)'
is
a
Blaschke
p oduc
o
e e y
a
E
D E
whe e
E
is
an
excep ional
se
o ze o
loga i hmic
capaci y
[5,
Theo em
11 .6
.4]
.
The
cons uc ions
o
Blaschke
p oduc s
in
[8],
[10]
i s
build
an
inne
unc ion
in
B
o
and
hen
apply
F os man's
heo em
.
S ephenson
asked
i
his
was
una oidable,
e
.g
.,
does
13 o
con ain
any
indes uc ible
Blaschke
p oduc s?
The
example
in
[3] is
buil
by
cons uc iog
he
ze o
se ,
so
does
no use
F os man's
heo em
.
Fu he mo e,
a
a ian
o
S ephenson's
cons uc ion
gi es
a
Blaschke
p oduc wi hou
using
F os man's
heo em
(see
nex
sec ion)
.
In
his
no e
we
expand
on
his
obse a ion
o
gi e
a
"cu
and
pas e"
cons uc ion o
an
indes uc ible
Blaschke
p oduc
in
130
.
INDESTRUCTIBLE
BLASCHKE
PRODUCT
IN
BO

9
7
One
could
also
y
o
p oduce
such
an
example
by
inding
a
su icien
condi ion
on
he
ze os
o
he
p oduc
o be
indes uc ible
and which
includes
some
sequences
sa is ying
he
13
o
condi ion
om
[3]
.
One
such
su icien
condi ion
o
indes uc ibili y
is
ha
he
sequence
be
hin,
Le
.,
zk
-
zn
kan
1
-
znzk
Howe e ,
his
condi ion
is
incompa ible
wi h
he
BO
condi ion
.
An
e en
mo e
ambi ious
p oblem
is
o
cha ac e ize
indes uc ibili y
in
e ms
o
he
ze o-se
.
In
[6],
Mo se
has
cons uc ed
a
des uc ible
Blaschke
p oduc
which
becomes
indes uc ible
when
a single
poin
is
dele ed
om
i s
ze o-se
.
This
indica es
any
cha ac e iza ion
o
indes uc ibil-
i y
in
e ms
o
he
ze o
se
would
be e y
delica e
(and
p obably
e y
di icul )
.
A
ela ed
p oblem
has
been
sol ed,
howe e
.
In
some
sense,
ini e
p oduc s
o
in e pola ing
Blaschke
p oduc s
a e he
"con o mally
in a ian "
class
o
Blaschke
p oduc s
.
In
[7]
Nicolau
has
gi en
a ze o-
se
cha ac e iza ion
o
hose Blaschke
p oduc s
B
so
ha
B
a
is
a
ini e
p oduc
o
in e pola ing
Blaschke p oduc s
o
e e y
a in
he
disk
.
Thus
he
has
sol ed he
con o mally
in a ian
e sion
o
he
p oblem
o
cha -
ac e izing
indes uc ibili y
.
Ou
cons uc ion
gi es a
Blaschke
p oduc
whose
singula
se
( he
accumula ion
se
o
he
ze os)
has
measu e
ze o
.
I
we
could
cons uc
an
example
whose
singula
se
was
he
en i e
ci cle,
his
unc ion
would
also
ha e
he
p ope y
ha
i s
ange
se ,
R( ,
a)
=
{w
:
he e
exis s
z
n
-
a,
(z
n
)
=
w},
equals
he
whose
disk
o
e e y
a
E
T
.
Ca mona
and
Cu í
had
asked
in
[4] i
he e
was
a
unc ion
in
H°°
n
BO
wi h
his
p ope y
.
I
belie e
he
cons uc ion
can
be
modi ied
o
gi e
such
an
example,
bu
a he
han
do
his,
I
will
ske ch
he
cons uc ion
o
a
unc ion
E
H°°
n
VMO
wi h
11
jj,,
=
1
and
R( ,
a)
=
D
o
e e y
aE
T
.
Since
VMO
C
BO,
his
is
an
e en
s onge
esul
(again
answe ing
a
ques ion
o
Ca mona
and
Cu í)
.
I
hank
A u o
Nicolau
o
b inging
his
ques ion o
my
a en ion
and
ou
discussions
on
i
.
I
also
hank
he
e e ee
o
his
help ul
commen s
.
His
sugges ions
ha e
cla i ied
he
exposi ion
in
se e al
places
.
98

C
.
J
.
BISHOP
2
.
The
B
O
cons uc ion
The
idea
is
qui e
simple
;
we
will
build
a
simply
connec ed
Riemann
su ace
by
aking
copies
o
he
uni disk
wi h
sli s
and
"gluing"
di e en
copies
along
he
sli s
.
A
simple
example
o his
idea
is
o
ake
in ini ely
many
copies
o
D [2,1),
and
iden i ying
he
" op"
edge
o
one
copy wi h
he
"bo om"
edge
o
he
nex
.
See Figu e
1
.
.
.(D
(D (D (D
...
Figu e
1
.
A
single
example
s~
Le
So be
he
ini ial
shee ,
which
we
also e e o
as
he
"ze o h
shee "
.
This
shee
con ains
a
poin
co esponding
o
ze o
in
he
uni disk
and
we
e e
o his
poin
as "0"
on
he
su ace
.
Le
S

,
be n h
s age
o his
cons uc ion
( he
union
o
copies
-n
o n)
and
S
=
UnSn
he
limi ing
su ace
.
Fo
each
o
he e
su aces
he
poin
"0" e e s
o
he
poin
0
on
So
.
In
he
es
o his
pape
we
shall
assume
ha
any
Riemann
mapping
o
he
uni disk
o
a
cons uc ed
su ace
like
S
n o
S
maps
0
in
he
disk
o
he
poin
0
on
he
su ace
.
Whene e
we
alk
abou
ha monic on
he
su ace
i is
he
push
o wa d
o
no malized
Lebesgue measu e on
he
ci cle
unde
such
a
Riemann
mapping,
Le
.,
ha monic
measu e
will
always
be
wi h
espec
o
he
poin
0
on
he
ze o h
shee
.
S
is
simply connec ed
so
he e
is
a
Riemann
mapping
ob
:
D
-->
S
and
he e
is
an
ob ious
holomo phic
p ojec ion
P
:
S
-3
D
.
We
claim
ha
F
=
P
o
D
mus
be
an
inne
unc ion
because
all
he
ha monic measu e
o
S
li es
on
he
pa
o
he
bounda y
abo e
he he
uni
ci cle
.
To
p o e
his
we
conside
S
n
and
show
ha
he
ha monic measu e
o
he
wo
adial
sli s
in
i s
bounda y
a e
O(ñ)
.
To
do
his
we
mapS

,
o
a
hal
in ini e
s ip
W
=
{(x, y)
:
-oo
<
x
<
0,
-(2n
- -
1)7
<
y
<
(2n
+
1)7 }
by
he
mapping
INDESTRUCTIBLE
BLASCHKEPRODUCT
IN
B
o

99
-
z
-->
log( z

2
)
1-2z
which
has
a well
de ined
b anch
on
S

, .
The
poin
0
on
he
su ace
is
mapped
o
-1/2
and
he
adial
edges
a e
mapped
o
he ho izon al
edges
o
he
s ip
.
S anda d
es ima es
(e .g
.,
map
he
s ip o
a
hal plane
ia
sin(z/i)
and
use he
Poisson
in eg al)
show
ha
he
ha monic measu e
o
he ho izon al
edges
o
he
s ip
wi h
espec
o
he
poin
-1/2
a e
app oxima ely
1/n
.
By
con o mal
in a iance
o
ha monic measu e
he
claim
abou
S
n
is
p o ed
.
Thus
he
ci cula
pa
o
OS
n
has
measu e
>_
1
-
C/n
.
Taking
n
-->
oo
we
see
ha
he
ci cula
pa
o
OS
has
ull
measu e,
Le,
¡Fl
=
1
a
.e
.
on
he
uni
ci cle
.
In
ac ,
F
mus
be
Blaschke
p oduc
.
To
see
his,
ecall
ha
a
unc ion
in
he
uni
ball o
HI(D)
is
a
Blaschke
p oduc
i
he
leas
ha monic
majo an
o
log
l
1
is
0
(e
.g
.,
[5,
Theo em
II .2
.4])
.
This
says ha
F
is
a
Blaschke
p oduc
i
0
is
he
leas
ha monic majo an
o
log
¡P(z)
1
on
S
.
Le
u be
he
leas
ha monic majo an
o
logjP(z)1
on
S
.
Then
u
es ic ed
o
S,,,
is
ha monic and
has
bounda y
alues
0 on
P
-1
(T)
and
>_
log
2
on
he
wo
adial
sli s
in
OS,
.
Thus
0
>_
u(0)
>_
2
log
2
.
Since
his
holds
o
any
n,
u(0)
=
0
( ecall
ha
he
"0" in
u(0)
e e s
o
he
designa ed
poin
on
he ze o h
shee )
.
Since
u
is
nonposi i e
his
implies
u
-
0 and
so
F
is
a
Blaschke
p oduc
.
Le
a
E
D
and
Ta(z)
=
(z
-
a)/(1
-
áz)
.
An
a gumen
like
he one
abo e shows shows
ha
i
a
:,¿
2
hen
F
a
=
Ta o
F
is
a
Blaschke
p oduc
.
Howe e ,
since
no
poin
o
S
co e s he
poin
{2
},
F2
is
ne e
ze o,
so
mus
be
a
singula
inne
unc ion
(in
ac ,
since
F
is
con inuous
excep
o
one
bounda y
poin ,
up
o
o a ions
i
mus
be
exp(A
i+z
),
o
some
A
>
0,
Le
.,
he
singula
inne
unc ion
co esponding
o
a
posi i e
poin
mass)
.
To
build
an
indes uc ible
Blaschke
p oduc
we
will
ha e
o
a y
he
cons uc ion,
adding
shee s
which
co e
he
omi ed
poin s
o
ea lie
gene a ions,
and
in pa icula , so
ha
he
leas
ha monic majo an
o
log
1To,(P(z))1
on
S
is
0
o
any
choice
o
a
E
D
.
This
says
ha
no
only
is
each
poin
co e ed
in ini ely o en,
bu
he e
is
some
sense
in
which
i is
" equen ly"
co e ed
.
To
ge
F
in o
he
li le
Bloch
space
imposes ano he
cons ain
:
gi en
any
e
>
0
only
ini ely
many
o
he
shee s
we
a ach
may
con ain
a
disk
o
adius
e
.
This
a ises
because
o a
geome ic
cha ac e iza ion
o
,Cio
due
o
S egenga
and
S ephenson
[9]
.
Fo
analy ic
on
D
and
a
E
D,
>
0
hey
de ine
SZ
a
( )
o
be
he
componen
o
-1
(D
(
(a),
))
con aining
a,
F
a
( )
= Og
a
( )
n
T
and
(a)
=
sup{
:
F
a
( )
=
0}
.
Then
E
Bo
i
(a)
=
o(1) as
¡al -->
1
.
In

10
0

C
.
J
.
BISHOP
pa icula ,
i
he
Riemann
su ace
o
is
ob ained
by
iden i ying
copies
o
D
along
sli s,
hen
he
endpoin s
o
any
such
sli
a e
in
he
ideal
bounda y
o
he
su ace
.
The e o e
will
be
in
he
li le
Bloch
space
i
o
e e y
E
>
0,
hese
endpoin s
o
pas ed
edges
a e
E-dense
in
D
o
all
bu
ini ely
many
shee s
.
We
will
induc i ely
cons uc
a
sequence
o
posi i e
numbe s
{E
n ,}
ending
o
ze o,
a
sequence
o
ini e
poin
se s
{E
,},
a
collec ion
o
adial
line
segmen s
T

,,
and
wo
sequences
o
in ege s
{gn},
{h

}
ending
o
in ini y
.
The
se s
{Ej,
{T

}
will
sa is y
(1)
Tn
C
T a+l,
EnC
En+1,
EnC
T

.
(2)
The
endpoin
o
each
segmen
in
T,,
is
in
E,,,
.
(3)
E

,/E

,+1
is
an
e en
in ege
.
(4)
Adjacen
poin s
o
&
en
a
segmen
o
T,,
a e
a
dis an e
E
,
om
each
o he
.
(5)
SupzED
dis (z,
E
n
,)
G
lOE

.
See
Figu e
2
.
An
"edge"
I o
T

deno es
a
subin e al
en
T

,
connec ing
wo
adjacen
poin s
o
E,,,
Le
.,
a
componen
o
T

, E
n
.
We
le
en
,
deno e
he
numbe
o
edges
in
T

, .
In
he
cons uc ion
below
each such
edge
will
be
ea ed
as
wo
sepa a e
pieces
o
he
bounda y
o
he
domain
R
n
=
D T,,
co esponding
o
i s
wo
sides
.
One
side
will
be
pas ed
e
a
shee
o
p e ious
gene a ion,
he
o he
pas ed
e
one
o
mo e
shee s
in
he
nex
highe gene a ion
.
Figu e
2
.
E

,
T

,,
R
n
INDESTRUCTIBLE
BLASCHKE
PRODUCT
IN
13p

10
1
Gi en an
edge
I
in
he
bounda y
o
R

,
we
can
ei he
a ach
ano he
copy
o
R
n
,,
o
di ide
he
edge
in o
m
=
En/En+1
edges
in
T
.
.+1 (
since
E
n
C
En+1)
and
a ach
m
copies
o
R,+1
.
Gi en
a
sequence
o
in ege s
{gi}
we
could
build
a
Riemann
su ace
as
ollows
.
S a
wi h
one
copy
o
R1
and
a ach
2e1
copies
o
o
R1
along (bo h
sides o )
each edge
o
T
1
.
Call
his
5
1
.
Then
a ach
mo e
copies
o
R1
along
each edge
in
OS,
o ob ain
S2and
con inuing
o
g
1
gene a ions,
ob aining
a
nes ed
sequence
o
su aces
S1
C
S
2
C
. .
.
C
Sgl
.
The
e m
"gene a ions"
e e s
o
he
ac
ha
o
connec
he
poin
0
in
he
ze o h
shee
So
o
any
o
he
unpas ed
edges
o
Sk
a
pa h
mus
pass
hough
a
leas
k
+
1di e en
shee s
(i .e
.,
copies
o
R1)
belonging
o So,
S1 So,
...
,
Sk Sk_1
.
We
ha e
ob ained Sgl
by
pas ing
oge he
iden ical
shee s,
Le
.,
copies
o
Rl
.
To
ge he
nex
su ace,
Sgl+1,
we
a ach
o
each
unpas ed
edge
o
Sgl
El/E2
copies
o
he
shee
R2
.
We
ob ain
Sgl+2
by
pas ing
a
copy
o
R2
o
each
unpas ed
edge
o
Ssl+1
.
We
con inue
in his
way
o
92
gene a ions,
ob aining
a
su ace
S91+92
.
The
nex
su ace
Sg1+g2+1,
is
cons uc ed
by
a aching
copies
o
R3
o
he
unpas ed
edges
o
Sg1+g2
.
Thus
gi en
he
sequence
o
in ege s
{gk}
(which
ells
us
o
how
many
gene a ions
o
a ach
copies
o
Rk)
and
con inuing
in
he
ob ious
manne ,
we
ob ain
an
inc easing,
nes ed
sequence
o
simply connec ed
su aces,
{S
n
}
.
Then
S
=
UnSn,
is
a
simply
connec ed connec ed
Riemann
su ace
.
I
-P
:
D
-
S
is
he
Riemann
map
(mapping
0
o
0 on
So),
and
P
:
S
-->
D
he p ojec ion
hen
F
=
P
o
<P is
a
holomo phic
unc ion
on
he
uni disk
which
we
claim
is
an
in ini e
Blaschke
p oduc
in
130,
i
he
pa ame e s
a e
chosen
co ec ly
.
This
is
essen ial
S ephenson's
cons uc ion
in
[10]
.
The
ac
ha
F
E
BO
ollows
om
he cha ac e iza ion
o
S egenga
and
S ephenson
men ioned
ea lie
.
I
he
sequence
{gi}
g ows
quickly
enough,
S ephen-
son
shows
he
mapping
F
is
an
inne
unc ion
.
I
dis (o,T
n
)
>_
En
hen
F
is
ac ually
a
Blaschke
p oduc
(again
i
gn
/
oo
as
enough)
.
To
p o e
his,
conside
he
leas
ha monic
majo an
u
o
log
IP(z)j
es ic ed
o
SN
=
Sgl+
.
.
.+g"
The
bounda y
b eaks
in o
wo
pieces
aS
N
=
01SN
U
a2SN
co esponding
espec i ely
o
P
-1
(T)
and
he
adial
edges
.
Then
u
has
bounda y
alues
0
on
8
1
SN
and
u
>_
109
En
on
á2SN
.
The
se
a2SN
can
be
made
o
ha e
as
small
ha monic measu e
as
we
wish
by
aking
gn
la ge
enough,
so
we
may
ake
0>-
U>
-
W(ó2SN)109E
n
>-
1
-
n
i
gn
is
la ge
enough
( ecall
ha
as be o e,
ha monic measu e
e e s
o
he
ha monic measu e
wi h
espec
o
he
poin
0 on
he
ze o h
shee
10
2

C
.
J
.
BISHOP
So)
.
Thus
F
is
a
Blaschke
p oduc ,
bu
i
canno
be
indes uc ible
since
i
only
akes
alues
in
each
E
n
ini ely
o en
.
As
S ephenson
poin s
ou ,
his
example shows
he
excep ional
se
in
F os man's
heo em
may
be
dense
in
he
uni disk
.
To
make
F
indes uc ible,
we
modi y
he
cons uc ion
sligh ly
.
Asso-
cia ed
o
each
E,,
de ine
ano he
se
F
n
,
by
eplacing
each
zE
E
nby a
poin
w
E
En,+1
wi h
Iz
-
w
i
=
á
E a
and
such
ha
w
is
on
same
adius
as z
.
The
se s
F

,
sa is y
app oxima ely
he
same
densi y
condi ions
as
he
E
n
(wi h
E
n
eplaced
by
2c,

,)
.
Ou
idea
is
o
modi y
he
cons uc-
ion
by
al e na ing
he use
o
he
se s
E

,
and
F

in
he
cons uc ion
.
Since
E

,
1
F
n
,
=
0 his
means
ou
su ace
will
co e
he
whole
disk
and
since
max(dis (z,
En),
dis (z,
Fn))
>_
c

,/4 o
e e y
poin
z
in
he
disk,
we
should
be
able
o
p o e
ou
unc ion
is
indes uc ible
by
es ima ing
ha monic
measu e
ei he
on
he
"E

,-shee s"
o
'T,,-shee s"
(depending
on
whe he
z
is
a
om
E

,
o
a
om
Fn)
.
Howe e
since E",
1
F,,
=
0,
we
need
some
u he
modi ica ions
o
o
able
o a ach an
"F

,-shee "
o
an
"
En
-
shee "
.
This
is
how
we
a ach
a
F

,-shee
o
an
E
n
,-shee
.
Conside
a
com-
ponen
in e al I o
T
n
wi h
endpoin s
in
E

, .
Le
T
n
,
be
he
ana-
logue
o
T

,
o
he
se
F

,
and
le
R,,
=
D Tn,
.
Assume
(wi h-
ou
loss
o
gene ali y)
ha
F

,
has
been
chosen
so
T

,
C
T

, .
Le
{ao,
al,
.
. . .
an}
=
I
n
E

,+1,
lis ed
in
o de
(e
.g
.,
ao,
a a
a e he
end-
poin s
o I)
.
Le
F

,j
=
F
n
U
{aj
,
a
j+
1}
.
Along
each
in e al
(a
j
,
aj+1)
a ach
a copy
o
R
n
.
To
his
shee
we
a ach
copies
o
Ñ,,
along
all
componen
in e als o
,, F
j
.
We
con inue
in his
way,
a aching
copies
o
R
n
along
in e als
o
Tn Fn,
excep
o
hose
shee s
eached
by
ei he
looping
a ound
a
j
o
a ound aj+1,
in
which
case
we
a e o ced
o a ach
copies
o
R
n
along
in e als
o
he
o m
T,, F
n
U{a
j
}
(o
Tn F

,U{aj+1})
.
Some
o
hese
iden i ica ions
a e
illus a ed
in
Figu e
3
.
Mo e
p ecisely,
Figu e
3
shows
egions
on
ou
shee s,
labeled
I,
II,
III,
IV
.
Shee
Iis
pas ed
o
shee
II
along
he
edge
[aj,
aj
+
1]
.
Shee
II
is
pas ed
o
shee
III
along
edge
[q,
aj
]
and
o
shee
IV
along
he
edge
[p, q],
whe e
p,
q
a e poin s
o
F,,,
adjacen o
aj
.
The
solid
and
do ed
cu es
illus a e
pa hs
om
shee
I
o
shee s
III
and
IV
espec i ely
which
(mus ) pass
h ough
shee
III
.
No ice
ha
he
poin
A
E
E
n
in
he
ideal
bounda y
o
shee
I is
co e ed
when
shee s
II
and
IV
a e
pas ed
along
[p, q]
.
Simila ly
ao
E
E
n
is
co e ed
when
II
is
pas ed
o
III
along
[q,
aj]
(assuming
j
0
0
;
o he wise
i
would be
co e ed
by
some
shee
a ached
o
shee
IV)
.
INDESTRUCTIBLE
BLASCHKE
PRODUCT
IN
B
O

103
4
4
1
4
aj
Figu e
3
.
Modi ica ions
o
co e
E
n
Suppose
we
ha e
al eady
cons uc ed
a
su ace
S

,
whose
bounda y
consis s
o
a cs
co e ing
T
o
edges
o
T,,
.
To
each
componen
in e al
I
o
T
,
,n En
we
a ach
copies
o
R

,
as
desc ibed
abo e
.
Do
his
o
g,+1
gene a ions
.
The
esul ing
shee s
co e
E
n
,
( he
only
shee s
which do
no
co e e e y poin
o
E
n
,
a e
hose a ached along
subin e als
o
he
o m
(ao,
al)
o
(an_I,
an)
in
he
cons uc ion abo e)
.
We
call
he
esul ing
su ace
S
n
, .
To
he
bounda y
o
S,
z
a ach
copies
o
R
n+
1
=
D Tn+1
along
componen
in e als
o
Tn+1 En+1
o
hn+1
gene a ions
( his
poses
no
di icul ies
since
E
n
,
F
n
and
all
poin s
o
he
o m
a
j
in
he
p e ious
s age
o
cons uc ion
we e
in
En+1
;
hus
e e y
adial
in e al
in
he
bounda y
o
S
n
has endpoin s
in
En+l)
.
The
esul ing
su ace
is
called
Sn+I
and
sa is ies
he
induc ion
hypo hesis
.
The
union
o e
n
o
hose
(nes ed)
su aces
is
deno ed
S
and
we
ob ain
he
desi ed
unc ion
by
mapping
he
disk
o
S
and
hen
p ojec ing
back
o
he
disk
.
All
ha
emains
is
o
choose
he
sequences
{En},
{gn}
and
{hn}
so
ha
he
ha monic measu e
es ima es
hold
.
We
will
i s
choose
gn
,
hen
En+1
and
hen
hn+1
Le
u
be
he
leas
ha monic
majo an
o
log
ITa
o
P(z)
1
.

We
wan