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P
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Ma
J
E
thema
tical
J
ISN
Physic
Electr
onic
Journal
1086-5
V
olume
P
ap
9,
er
203
4
Reciv
ed:
Sep
Editor:
16,
R.
de
20,
la
Hilb
Revisd:
Lla
v
Sep
10,
203,
Aceptd:
Sep
18,
203
ert's
e
Space
Filng
Curv
es
and
Geo
desic
Lamintos
V
ctor
F.
Septm
b
Sirv
er
en
10,
t
203
Abstrac
W
e
this
presn
e
helps
to
es
an
w
to
en
space
9.
aply
].
Hilb
ert's
ts
togehr
of
es
space
with
the
in
from
terv
in
ling
a
al
the
28A0,
has
curv
transv
are
terv
ersal
map
al
to
the
e
and
w
measur.
ed
to
26A7,
the
e
aso
ciate
The
the
square.
regula
W
n
to
laminto
e
genraliz
-gon.
53C2.
a
9
diern
t
laminto
desic
],
con
b
geo
the
and
helps
us
1890.
[5
the
n
Hilb
um
of
in
[1
]
um
understa
the
disc
the
ers
aso
e
of
]
is
space
geomtry
and
the
the
en
in
b
curv
space
w
,
dynamics
as
in
].
curv
in
in
v
es
ha
olv
tro
duce.
In
e
[13
]
den
ed.
V.F
V
enzula.
SIR
VENT,
e-mail:
Departmn
vsir
to
en
t@usb.v
de
Matem
atics,
Univ
ersida
e
1
Sim
on
Bol
v
ar,
Apartdo
890,
Carcs
the
v
of
14
ling
and
orde
history
e
[13
].
tegr
4
the
the
in
e.
in
curv
ed
as
[1
base
w
w
erg
some
ling
ling
to
ho
kno
space
e
curv
using
to
later
ciated
Sc
al
describ
studie
],
done
,
curv
ling
terv
used
tly
[10
en
as
this
space
[12
in
w
are
recn
yp
b
of
a
the
e
b
In
t
nski
curv
n
has
wing
of
on
More
it
dra
Sierp
ers
ert's
the
another
And
],
b
withn.
viour.
The
exampl
esgu
plane.
on
to
Leb
refncs
laminto
in
sytem.
tex,
eha
y
taion
functio
10
]
another
of
repsn
,
dynamicl
[6
taion
iterad
,
eano
published
b
repsn
the
the
P
[3]
others
on
of
y
ert
among
[8
h
b
Hilb
sytem
In
ric
duce
ear
duce,
of
es.
a
tro
y
digts
a
This
in
on
others
construed
[1
oin
curv
as
tro
maps
curv
e
author
p
wing
in
The
among
curv
w
based
the
ling
This
the
functio
base
studie
in
er
iterad
in
h
e
fol
are
and
b
curv
exampls
whic
clasi
disk
(20):
the
curv
eano's
the
the
ling
clasiton
During
thes
P
of
w
space
ling
[7].
other
base
ersion
on
ho
ject
space
Later
v
laminto
duction
rst
in
die
to
tro
wn
Al
mo
desic
understa
sub
In
sho
a
geo
construi
AMS
The
a
us
this
1
t
curv
1086-A,
the
e
W
b
y
e
w
geo
ould
lik
desic
exampls
it
pro
to
p
w
to
e
giv
en
y
laminto
p
aso
e
a
t
ciate
a
a
in
geo
desic
ers
D
.
h
space
curv
with
thes
curv
pro
the
e.
ert
y
W
.
of
the
square,
e
This
geomtry
of
Theorm
joined
clasi
e
desir
understa
in
ts
the
ling
symetri
sumarized
oin
In
ling
the
to
p
e.
space
with
the
tha
curv
clasi
us
are
are
the
disk
helps
suc
ling
ert's
the
h
e
space
Hilb
laminto
result
curv
the
to
die
on
the
ling
y
laminto
mo
whic
The
space
b
desic
e
measur,
of
a
plane
laminto
symetri
group
the
w
geo
transv
The
diheral
to
oin
Therfo
curv
e.
the
desic
same
to
new
curv
b
osible
with
ling
geo
the
explain.
comes
space
a
to
shal
this
laminto
ciate
ed
not
as
ciate
aso
map
is
ertis
aso
e
are
the
i.e
are
1.
4
Geo
desic
lamintos
refncs
on
withn,
geo
desic
4
diern
t
diern
t
one.
w
ev
only
whic
it
has
The
few
In
last
ther
are
2
case
some
W
e
shal
e
Denito
e
X
The
to
con
],
caled
vide
atr
f
;
H
;
H
;
1
(
z
f
h
;
h
;
0
h
;
1
the
ontr
)
h
and
,
15
]
are
e.
W
ertis
as
thes
t
w
o
visted
e
obtain
the
a
previous
lamintos
are
higer
dimenso.
es
from
the
in
terv
simlar
al
pro
to
the
regula
ertis.
Ho
w
ev
er
d.
curv
clasi
e
space
ling
curv
e,
via
iterad
functio
sytem
or
IFS
c
onsit
of
a
c
omplet
metric
mapings.
on
the
space
her
of
the
g
b
e
an
IFS
on
R
=
f
x
al
compat
subet
induce
+
i
z
=
;
H
(
z
)
map
iy
2
C
i
=
j
0
is
z
+
;
H
(
1
g
2
of
a
x;
y
con
b
e
an
z
)
1+
=
X
.
W
tracion.
e
consider
Its
1
g
,
this
xed
p
oin
t
is
wher
i
z
+
;
H
(
2
2
IFS
on
I
=
[0
2
;
1],
z
)
i
i
=
+
+
1
:
3
2
wher
h
(
3
forme
o
ert's
action
map
2
of
square
curv
pro
with
functio
c
a
0
Let
]
3
z
H
the
die
tialy
curv
and
[4
[13
IFS.
H
2
tha
same
esn
In
In
square.
ling
en
texs.
set.
dynamics.
mo
lamintos
ev
olic
so
to
space
ert's
metric,
the
,
b
ert's
genralizd
desic
iterad
of
Hausdor
of
0
n
set
induce
actor
H
A
nite
the
y
n
Hilb
sym
the
the
con
].
80)
a
with
the
Let
page
a
e
Hilb
of
of
Hilb
tha
on
t
Julia
hanged,
in
e
b
w
the
c
k-wise
explain
can
case
e
is
c
curv
geo
the
yp
diern
quadrtic
sharing
of
simlar
en
[10
mapings
pro
w
in
with
tracion
space
et
square
clo
also
constru
of
done
([2
gethr
a
also
construi
is
2.1
ac
er
t
laminto,
e
of
some
the
e
yp
pa
ersion
the
as
t
this
in
can
b
describ
(IFS),
sp
in
den,
e
v
sytem
W
this
study
of
ti
t
in
the
dels
diern
to
dierncs
Revisd
a
previously
in
an
symetri.
e
w
studie
used
or
get
wn
w
this
en
on
k-wise
ciated
section
In
b
mo
c
e
er
sho
the
-gon.
w
aso
construi
e
geomtric
clo
h
lamintos
v
sytem
the
from
er
as
functio
from
e,
ha
are
up
iterad
orde
curv
Ho
the
the
disk
lamintos
turn
section
a
n
desic
lamintos
In
in
geo
the
2
t
)
=
2
t=
4
+
k
=
4,
for
0
2
k
3.
The
atrco
k
IFS
is
the
square
R
and
of
the
later
is
the
in
terv
al
I
.
This
IFS
on
the
in
terv
al
P
1
n
dens
the
n
umeration
sytem
base
4,
i.e
t
=
a
=
4
,
with
0
a
n
n
t
=
lim
h
n
!1
h
(
a
if
and
only
if
n
=1
).
a
1
The
I
3
space
n
ling
curv
e
:
I
!
R
is
den
as
H
(
t
)
=
lim
H
H
H
a
H
a
(
R
)
;
a
1
n
2
n
!1
P
1
n
wher
t
=
a
=
4
.
This
map
is
con
tin
uos
and
surjectiv
e.
n
n
W
e
wil
=1
consider
a
mo
die
v
ersion
of
this
curv
e:
Let
f
H
;
H
;
0
f
h
0
;
h
;
h
1
;
h
2
g
b
e
an
IFS
on
I
=
[0
;
1],
H
;
1
H
g
2
b
e
an
IFS
on
R
3
wher
3
z
H
(
z
)
1+
=
i
+
iz
;
0
H
(
z
)
1+
=
i
+
z
;
H
1
2
2
(
z
)
1+
=
i
+
2
H
(
z
)
=
i
1+
i
+
:
3
2
2
z
;
2
2
2
2
2
and
Figure
1:
The
1
clasi
Hilb
ert's
curv
e.
12
13
21
11
10
20
22
2
03
0
23
00
30
01
33
31
3
02
Figure
2:
The
mo
die
Hilb
ert's
curv
32
e.
And
8
8
t
7
1
+
if
0
t
t
(
t
)
32
8
=
h
t
1
(
t
)
t
<
1
23
t
)
if
0
t
h
h
(
t
)
=
h
(
k
<
1
if
0
(
t
)
32
8
=
15
1
t
)
t
if
32
+
t
<
1
k
=
4
for
k
23
;
thes
if
4
=
1
;
2
;
3.
W
e
wil
denot
32
b
y
t
<
1
:
8
I
=
[
k
=
4
;
(
k
+
1)
=
4)
=
h
(
k
are
space
1
+
8
0
3,
the
t
;
:
+
And
1
3
t
;
<
31
+
4
=
4
2
t
<
8
:
;
8
t
<
32
2
1
if
32
1
+
(
1
+
4
8
t
4
h
7
;
8
<
tha
<
8
t
32
8
;
t
=
1
if
0
:
4
=
0
32
1
:
k
if
4
0
Note
1
+
<
4
h
15
<
<
the
in
ling
terv
als
curv
e
den
b
:
I
y
!
the
R
I
),
for
k
IFS.
is
den
in
a
simlar
w
a
y
,
as
the
clasi
curv
e:
M
(
t
)
=
lim
H
M
n
H
!1
a
con
tracion,
t
H
(
),
wher
t
=
lim
h
n
2
and
R
a
1
a
a
(
h
!1
n
t
)
are
a
el
den.
In
a
simlar
w
a
h
(
).
Since
eac
h
h
and
H
is
k
2
y
I
a
1
w
a
k
n
to
the
clasi
case
w
e
can
pro
v
e
tha
M
is
con
tin
uos
and
surjectiv
e.
F
urthemo,
it
can
b
e
pro
v
ed
tha
it
is
H
older
con
tin
uos,
with
M
exp
one
t
One
1
=
of
2.
the
main
dierncs
in
thes
t
w
o
curv
es
is
tha
in
the
clasi
v
ersion
(0)
=
0
and
H
(1)
=
1.
And
in
the
revisd
v
ersion
the
images
of
the
p
oin
ts
0,
1
=
4,
1
=
2,
3
=
4,
1,
i.e
the
extrmis
the
cen
of
H
in
tre
terv
als
of
3
tha
the
den
square
Geo
the
R
IFS
on
the
in
terv
al,
are
desic
laminto
on
the
b
e
the
the
IFS:
f
p
oin
t:
(1
+
i
)
=
2,
1
D
Since
same
closed
unit
image
h
;
0
h
of
;
1
h
;
2
h
0
g
disk
and
as
the
geomtrical
disk
2
Let
the
.
acting
1
in
are
on
the
the
plane,
the
same
b
under
and
oundary
S
1
the
its
maps
b
of
the
3
3
disk.
of
oundary
the
.
IFS
on
W
e
the
iden
in
tify
terv
S
al.
with
W
I
e
think
=
[0
of
;
1).
the
Figure
The
in
construi
terv
of
als
den
b
y
the
3:
geo
the
desic
IFS,
The
geo
desic
laminto
i.e
laminto
t
=
k
=
4
is
with
.
as
k
fol
=
ws:
0
;
1
;
W
2
;
e
3.
consider
the
And
w
e
extrmis
join
of
pairwse
the
conseutiv
e
k
extrmis,
i.e
w
e
join
t
with
t
,
k
wher
j
=
k
+
1
(mo
d
4)
for
0
k
3,
b
y
arcs
of
cirles
tha
j
1
met
the
b
ther.
oundary
of
Therfo
Let
w
a
:
:
:
a
e
b
1
S
p
wil
e
erp
en
cal
a
w
ticulary
.
thes
ord
in
arcs
the
If
w
geo
e
think
in
the
h
yp
erb
olic
disk,
thes
arcs
are
geo
desic
alphb
desic.
et
f
0
;
1
;
2
;
3
g
.
W
e
join
b
y
a
geo
desic
the
p
oin
t
h
n
h
(
a
h
h
(
a
t
),
a
wher
j
=
k
+
1
(mo
d
4)
for
k
=
0
;
1
;
2
;
3.
W
e
do
this
for
al
p
t
)
a
k
1
with
n
osible
w
ords
j
1
n
2
in
this
alphb
et
and
later
w
e
tak
e
the
closure
in
the
Hausdor
top
olgy
of
2
D
.
The
elmn
ts
of
D
1
are
eithr
geo
desic
or
p
oin
ts
in
S
.
In
the
later
case
the
p
oin
ts
are
caled
de
genr
ate
ge
o
desic
.
2
Denito
3.1
tha
any
A
two
of
geo
desic
thes
laminto
ge
o
on
desic
do
not
D
is
inters
a
ct
no-empty
close
exc
ept
at
d
their
set
end
p
of
ge
o
desic
of
the
disk
and
oints.
2
Prop
ositn
Pro
3.1
of:
Let
a
is
a
terscion
ge
o
and
1
in
a
b
et
w
laminto
m
b
desic
b
b
1
en
the
on
e
t
w
o
w
D
ords
.
in
the
alphb
et
f
0
;
1
;
2
;
3
g
.
Sup
ose
tha
ther
is
an
l
geo
desic
tha
join
the
images
of
the
t
's
under
h
k
h
and
a
h
a
h
.
b
1
b
m
1
l
Therfo
the
in
terios
of
h
h
h
a
(
a
I
)
and
h
a
1
h
h
b
m
m
+1
(
b
l
some
0
a
;
m
is
a
sub-w
b
+1
l
ord
of
3.
This
is
not
p
osible,
unles
l
ositn
)
ha
v
e
no-empt
y
one
of
the
w
ords:
a
:
:
:
a
since
the
in
3.2
The
laminto
terios
terscion,
for
a
1
other,
in
+1
+1
the
of
h
(
I
)
and
h
(
k
Prop
I
b
1
is
invart
I
)
are
,
m
disjon
t
m
for
k
b
:
+1
:
:
b
b
1
6=
j
.
l
l
+1
j
under
the
gr
oup
of
symetri
of
the
squar
e
D
.
4
k
Pro
of:
Let
R
:
I
!
I
b
e
the
rotain
b
y
1
=
4.
Since
h
(
t
)
=
R
(
h
(
k
1
=
1
tha
for
an
y
t
and
an
y
w
ord
a
j
a
:
1
R
(
h
m
h
(
a
1
=
d
R
4).
So
b
y
the
construi
of
the
t
a
)
=
=
),
for
0
k
3.
W
e
ha
v
e
h
4
h
j
b
h
(
a
t
)
a
wher
b
=
j
a
+
1
1
1
4
1
(mo
t
0
4
m
laminto,
it
1
fol
ws
2
tha
m
is
in
v
arin
t
under
the
rotain
.
1
=
4
The
symetr
relatd
to
the
in
v
arince
under
the
0
to
the
fact
if
t;
rection
along
the
horizn
tal
edg
is
equiv
alen
t
0
t
are
joined
then
1
t
,
1
t
are
also
joined.
It
fol
ws
from
h
(
t
k
)
=
1
h
(
j
3
t
)
k
r
0
in
for
the
j;
v
k
ertical
3,
wher
axis
r
(0)
comes
=
from
1,
the
r
(1)
=
fact:
0,
r
(2)
h
(
=
k
t
)
3
=
and
(2
r
+
k
j
)
(3)
=
=
4
2.
The
h
in
(
k
in
v
arince
under
the
rection
in
the
diagonls
4
is
obtained
t
)
arince
under
for
k
=
the
0
;
+1
r
The
v
in
a
(
simlar
j
)
w
a
y
.
2
and
(
j
0
rection
)
j
3.
1
Æ
2
0
0
b
b
b
b
1
2
2
Figure
4:
The
1
construi
of
C
.
Æ
0
Prop
ositn
3.
L
et
b
e
an
elmnt
of
with
end
p
oints
b
and
0
b
.
Then
(
b
)
=
(
M
Pro
of:
By
the
construi
of
ther
are
geo
desic
suc
h
tha
they
b
)
.
M
join
the
images
under
the
k
IFS
of
the
t
's.
And
they
con
v
erg
to
.
k
0
If
the
end
p
oin
ts
of
are
b
and
k
0
b
then
(
k
b
)
M
=
(
k
b
).
So
b
y
the
con
tin
uit
y
of
the
space
M
k
k
0
ling
curv
e:
(
b
)
=
(
M
Ho
H
w
(
R
ev
)
er
con
v
H
(
k
k
so
they
R
ers
of
),
Prop
ositn
diern
t
3.
from
the
is
cen
not
true.
tre
of
If
R
:
(1
w
+
e
i
)
tak
=
e
2,
a
p
then
oin
t
its
in
the
b
oundary
preimags
b
lie
in
I
w
en
I
k
b
pro
et
and
+1
canot
Thes
).
M
the
and
b
e
k
+1
joined.
ertis
alo
w
us
to
den
a
map
:
!
R
as
fol
ws:
Let
b
e
a
geo
desic
of
with
0
end
p
oin
ts
b
,
b
.
So
(
)
:=
(
b
)
:
By
Prop
ositn
3.
this
map
is
w
el
den.
The
map
is
M
con
tin
uos
and
surjectiv
e
since
has
thes
pro
ertis.
M
Remark:
In
are
map
the
ed
IFS,
the
to
clasi
the
i.e
case
cen
I
tral
with
p
j
=
0
w
oin
;
t
1
;
2
e
of
;
do
not
the
3.
end
up
square
So
with
are
if
thes
a
not
p
oin
b
ts
laminto
b
oundary
p
are
joined
oin
ecaus
ts
and
w
the
of
e
the
four
in
p
itera
terv
oin
als
the
ts
tha
tha
pro
den
ces,
w
e
end
j
up
with
crosing
curv
es:
in
P
3.1
eano's,
the
Leb
The
geo
desic.
This
esgu',
situaon
is
also
found
in
the
other
clasi
space
ling
etc.
transv
ers
measur
to
the
laminto
2
Let
Æ
geo
desic
geo
desic
sa
b
e
an
y
to
arc
w
in
ards
C
.
joing
the
can
y
D
b
e
More
t
b
orien
w
o
oundary
distnc
geo
of
ted.
This
precisly:
the
pro
Let
giv
b
1
of
the
acording
cedur
and
Æ
desic
disk
e
laminto.
the
es
rise
the
to
geo
It
t
a
w
o
Can
p
desic
tor
in
osible
set
can
b
cedur
dens
t
w
o
disjon
t
slid
along
in
tha
in
the
are
b
the
whic
oundary
h
of
joined
b
y
an
the
the
disk,
arc
Æ
.
This
2
0
pro
e
directons
in
terv
als
on
the
cirle
J
=
[
b
;
b
]
1
and
0
J
=
[
0
b
;
0
b
]
wher
b
;
2
b
are
k
2
1
k
the
end
p
oin
ts
of
for
k
=
1
;
2.
Let
b
e
a
geo
desic
in
the
laminto
suc
h
tha
its
end
p
oin
ts
lie
k
0
on
the
same
p
oin
ts
p
oin
ts
in
of
.
terv
al
Se
J
or
0
J
gure
4.
,
w
e
The
remo
v
set
e
C
from
is
J
[
J
obtained
the
op
in
en
this
w
in
a
terv
y
al
when
whose
extrmis
al
the
are
geo
desic
in
the
end
with
end
Æ
0
in
Let
J
Æ
or
b
J
e
an
are
y
consider.
transv
ers
arc
to
.
W
e
den
(
Æ
)
=
M
(
C
)
s
wher
M
is
Æ
and
s
is
the
Hausdor
dimenso
of
C
ositn
s
-Hausdor
0
0
.
0
Prop
the
s
0
measur
Æ
3.4
F
or
evry
tr
ansver
curve
Æ
to
,
the
Hausdor
dimenso
of
C
is
s
=
Æ
Befor
pro
ving
this
pro
ositn
w
e
ned
to
describ
e
the
struce
of
the
set
of
the
1
=
2
form
h
h
a
ho
w
h
h
h
a
h
h
a
(
I
)
for
I
)
ts
in
the
previous
set.
W
e
cal
cylinders
al
a
n
cylinders,
:
:
:
a
1
w
set
of
the
+1
a
w
ord
in
the
alphb
et
f
0
;
1
;
2
;
3
g
.
In
orde
to
understa
n
n
the
the
a
n
a
1
of
(
a
1
e
giv
e
the
fol
wing
mo
del
for
them:
Let
0
x
<
1
y
<
1
y
<
2
x
x
2
<
5
3
y
<
3
y
<
4
x
4
1
the
(
a
1
and
.
0
struce
n
form
I
),
y
y
y
y
2
3
4
1
x
x
x
=
x
1
4
2
y
3
y
y
1
y
2
3
4
x
x
x
x
1
2
Figure
with
the
5:
3
Diern
t
t
4
yp
e
of
cylinders.
relations
j
y
y
j
2
1
j
x
x
j
2
+
j
x
y
j
The
extrmis
of
+
j
j
1
x
1
the
j
aso
x
j
1
3
x
1
y
4
x
j
+
j
y
j
+
j
x
4
3
x
1
j
x
j
4
x
2
ciated
;
j
1
4
=
4
3
cylinder
x
;
j
4
y
j
3
1
x
j
+
x
2
=
2
x
3
j
+
j
=
4
3
y
x
j
2
;
j
4
4
x
y
2
x
1
j
j
x
1
=
j
to
an
x
1
y
w
4
ord,
sa
:
j
4
3
y
a
:
:
:
a
,
1
And
the
y
's
wil
b
e
the
extrmis
of
its
sub
cylinders,
i.e
the
wil
b
e
the
p
oin
ts
x
's.
n
j
cylinders
aso
ciated
to
the
w
ords
j
a
:
:
:
a
a
1
.
n
gure
n
5.
Let
If
[
x
;
x
)
1
=
t
x
w
2
If
consider
are
x
us
+1
Ther
w
e
o
sa
p
y
x
<
x
,
w
e
with
tha
the
;
x
),
3
cylinder
w
e
wil
of
t
yp
e
sa
y
the
tha
in
cylinder
:
:
:
the
terv
cylinder
als
coresp
a
j
1
the
cylinders
are
of
this
form.
Se
is
of
t
yp
e
2.
I
,
I
,
I
,
2
I
and
3
using
inducto
one
can
pro
v
e
the
fol
wing
fact
4
is
of
t
ondig
yp
e
1
to
if
j
=
1
;
the
2
;
w
3
ord
and
a
is
:
of
:
:
a
is
of
t
yp
e
1
then
the
cylinder
coresp
ondig
n
t
yp
e
2
if
j
=
0.
And
if
the
cylinder
coresp
ondig
to
n
w
ord
a
:
:
:
a
is
1
and
is
of
Prop
t
yp
e
2
if
j
0
al
of:
yp
e
2
then
the
l
al
thes
By
;
cylinder
coresp
ondig
ge
o
joing
l
the
is
obtaine
cylinders
d
and
by
then
joing
by
taking
the
x
a
:
:
:
a
j
is
of
t
yp
e
1
if
j
=
1
;
3
n
ge
o
desic
closur
the
e,
in
neighb
the
ouring
extr
Hausdor
top
emits
ol
gy,
of
the
desic.
neigh
b
ouring
extrmis
of
[
x
;
x
)
1
with
to
2.
laminto
for
of
Pro
=
The
cylinders,
unio
t
1
3.5
the
of
n
ositn
of
the
cylinders:
the
a
al
1.
1
to
se
4
is
1
the
If
x
3
Staring
out
[
3
2
ab
[
2
osiblte:
[
[
x
;
2
x
)
3
w
e
mean
joing
x
with
4
x
and
2
x
3
1
.
4
Let
h
h
(
a
I
)
b
e
a
cylinder.
Its
extrmis
are
of
the
k
eithr
h
3,
or
h
(
a
h
(
a
)
for
some
k
1
h
t
a
n
~
form:
a
1
0
n
1
~
t
),
for
some
1
l
n
,
wher
t
=
1
=
8,
the
discon
tin
uit
y
p
oin
t
of
the
maps
a
1
l
~
h
:
I
!
I
;
note
h
(
k
t
)
=
t
.
k
t
's.
The
neigh
b
Since
the
neigh
b
ouring
extrmis
coresp
ond
to
images
of
conseutiv
e
k
ouring
extrmis
of
a
cylinder
are
joined
b
y
geo
desic,
acording
to
the
denito
k
of
the
laminto
.
On
the
other
hand.
Giv
en
h
h
(
b
oin
ts
are
neigh
b
ouring
t
)
b
and
h
k
1
p
of
h
(
b
cylinder
h
h
of
of
geo
is
Prop
ositn
desic
of
t
yp
3.4:
tha
e
2.
join
So
Withou
los
extrm
the
p
cylinder
is
oin
of
of
ts
of
the
genralit
the
[
x
;
y
)
1
of
the
limt
[
[
y
;
1
set
x
),
4
C
[
y
;
4
the
x
form
)
2
in
terv
C
is
[
[
x
als
[
y
;
y
as
in
y
x
;
),
x
[
)
;
y
;
3
x
),
4
of
[
)
x
;
x
K
=
K
2
[
K
K
1
k
+
1
(mo
d
4).
Thes
tha
e
whic
the
h
can
end
is
sup
sub
ose
p
oin
ts
tha
dive
of
this
in
to
[
x
;
y
1)
y
remo
;
v
C
[
are
y
).
In
the
rst
step
of
the
the
fol
wing
construi
4
ed
(se
gure
6).
4
cylinders:
[
Æ
cylinder
=
\
K
j
,
0
wher
in
this
case
K
j
=
[
y
;
4
x
)
and
K
[
y
;
2
6
1
;
1
x
)
[
2
1
=
4
1
x
0
2
=
1
1
ose
w
),
[
1
,
1
sup
3
are
=
4
and
Æ
1
x
[
y
Æ
[
can
2
)
3
e
3
;
1
y
2
terscion
y
).
and
2
[
3
),
1
obtained
;
3
Æ
So
[
2
w
cylinder,
1
cylinders:
j
I
k
n
y
same
(
b
1
on
with
n
h
b
Pro
),
j
1
the
t
b
n
extrmis
x
)
2
[
[
x
;
3
y
).
Note
tha
K
2
and
K
3
1
1
Æ
y
y
y
x
x
1
y
x
x
2
3
1
2
Figure
6:
4
3
The
4
construi
of
C
.
Æ
1
(
K
)
M
1
2
(
K
)
M
1
Figure
7:
The
image
of
a
cylinder
under
and
its
sub-divon.
M
are
cylinders
of
t
yp
e
2.
W
e
con
tin
ue
the
sub
divson
of
thes
cylinders
in
the
same
w
a
y
as
for
the
j
paren
t
cylinder.
This
pro
ces
can
b
e
describ
ed
in
term
of
an
IFS,
f
'
;
'
g
1
suc
h
tha
K
=
'
(
2
K
)
j
0
1
for
j
=
1
;
2.
This
IFS
satie
the
op
en
set
condit.
Since
eac
h
'
has
a
con
tracion
factor
of
1
=
4,
j
the
resulting
tec
Can
hniques
tor
tha
set
0
<
has
Hausdor
M
(
dimenso
C
)
<
1
.
1
=
2.
More
v
er
it
can
b
e
pro
v
ed
using
standr
Æ
1
Prop
ositn
3.6
L
et
=
2
Æ
b
e
any
ar
c
tr
ansverl
to
whose
end
p
oints
ar
e
in
the
ge
o
desic
and
1
.
The
image
of
the
set
C
under
2
is
the
line
se
gment
tha
joins
(
)
Æ
Pro
of:
geo
Withou
los
desic
tha
of
join
and
(
)
1
genralit
y
extrm
p
oin
ts
w
e
of
can
a
sup
ose
cylinder
tha
of
t
yp
the
e
2.
end
Sa
y
[
p
x
;
oin
x
)
1
.
2
ts
[
[
of
x
the
;
2
x
arc
).
3
Æ
Se
are
in
gure
the
6.
As
w
e
4
1
sho
w
ed
in
the
pro
of
of
Prop
ositn
3.4,
C
=
\
K
Æ
j
wher
0
K
=
j
[
x
;
0
x
)
1
[
[
x
;
2
x
),
3
K
=
4
2
K
[
K
,
1
1
1
2
K
=
[
x
;
y
1)
[
[
y
;
1
x
]
4
and
K
=
[
y
;
4
x
)
2
1
[
[
x
;
2
y
).
3
The
cylinder
K
is
3
map
ed
b
y
in
0
to
a
M
1
sub-qare
of
R
.
By
the
denito
of
the
space
ling
curv
e
(
x
)
M
of
this
cylinder
[
x
;
x
)
1
x
and
(
1
x
)
M
are
op
osite
cornes
2
sub-qare.
The
[
1
;
y
),
3
[
y
;
3
y
)
1
sub-qare
[
[
[
(
;
K
;
y
)
is
sub
dive
in
to
the
sub-cylinder:
[
x
;
4
).
y
)
1
And
the
image
under
of
4
).
M
x
3
y
3
of
x
2
and
2
eac
h
of
thes
[
[
y
;
1
x
),
4
cylinders
[
y
;
4
x
)
2
coresp
[
2
onds
to
a
M
Al
thes
square
ha
v
e
the
same
are
and
they
in
tersc
in
only
one
p
oin
t,
the
0
n
cen
tre
of
(
K
).
M
So
(
0
K
)
M
is
the
coletin
of
t
w
o
of
thes
four
sub-qare.
Then
(
1
\
K
j
j
is
a
coletin
of
smal
square
whose
diagonl
are
in
the
line
segmn
t
tha
joins
(
x
)
M
So
in
the
limt
w
e
get
(
\
K
M
Let
F
:
!
b
e
j
a
)
0
is
the
line
segmn
t
tha
joins
with
end
p
map
oin
b
x
)
M
on
ts
(
j
the
laminto,
den
as
fol
0
laminto
)
M
and
and
(
1
x
Let
=0
(
x
).
M
).
M
ws.
and
1
2
b
2
e
a
geo
desic
on
the
0
b
(if
is
degnrat
b
=
b
).
So
the
image
of
under
F
is
the
0
geo
desic
tha
joins
f
(
b
)
and
f
(
b
)
wher
f
is
the
expandig
map
on
the
in
terv
al
den
b
y
the
1
in
v
ers
of
the
maps
in
the
IFS,
i.e
f
(
t
)
=
h
(
t
)
if
t
2
I
.
The
con
tin
uit
y
of
F
fol
ws
from
the
k
k
con
tin
uit
y
of
f
.
On
the
other
hand
if
2
is
suc
h
tha
it
joins
h
h
(
a
for
n
2
and
j
=
k
+
1
(mo
d
4),
then
F
(
)
joins
h
h
(
a
F
F
then
()
The
(
)
=
1
=
8,
and
it
can
b
e
easily
c
hec
k
ed
tha
t
p
oin
t
with
h
h
(
a
a
h
(
t
),
a
t
j
n
).
a
If
n
=
1
j
2
is
a
h
k
n
degnrat
geo
desic.
Therfo
.
domain
of
F
can
b
e
extnd
to
the
set
of
equiv
alenc
clase
of
transv
0
laminto
and
1
n
this
)
k
n
)
a
2
t
a
1
.
Giv
en
Æ
and
Æ
ers
curv
es
to
the
0
t
w
o
transv
ers
curv
es
7
to
,
w
e
sa
y
tha
Æ
Æ
if
the
end
p
oin
ts
of
eac
h
0
0
curv
e
lie
in
the
same
pair
of
distnc
geo
desic
and
C
=
C
.
Æ
the
denito
F
(
Æ
is
of
)
is
the
den
as
clear
tha
Prop
map
a
this
F
curv
to
e
The
ersal
ersal
is
3.7
transv
transv
denito
ositn
the
only
to
extnd
map
curv
to
F
has
the
al
F
the
pr
es
to
(
op
)
equiv
and
F
their
wher
alenc
erty
Therfo
equiv
are
Æ
)
=
=
2
alenc
the
clase
(
(
Æ
).
W
e
extnd
Æ
geo
of
clase.
desic
The
transv
transv
ersal
curv
ersal
curv
e
to
Æ
.
It
es.
.
1
1
Pro
of:
By
denito
F
(
(
Æ
)
=
(
F
(
Æ
)
=
M
(
C
).
On
the
other
hand
the
Can
tor
set
s
F
(
Æ
)
0
C
is
of
the
form
\
K
Æ
j
wher
0
K
is
j
a
nite
unio
of
closed
in
terv
als,
as
sho
wn
in
the
pro
of
of
j
1
1
1
1
Prop
ositn
3.4
So
f
(
C
)
=
\
f
Æ
j
(
K
)
0
and
C
=
f
(
C
).
j
Since
f
is
4
to
1
and
eac
h
of
Æ
F
(
Æ
)
1
the
branc
hes
of
f
,
i.e
the
maps
h
's,
is
a
con
tracion
with
a
factor
of
1
=
4,
w
e
ha
v
e
k
3
3
[
X
1
1
M
(
C
)
=
M
(
s
f
(
C
)
s
F
(
Æ
=
M
(
Æ
h
C
)
k
0
=
M
(
Æ
h
(
s
C
)
k
0
=
Æ
0
k
1
(
s
)
0
=0
k
=0
s
0
=
4
M
(
C
)
s
=
2
M
(
Æ
Henc
F
=
2
.
C
)
s
=
2
(
Æ
)
:
Æ
0
0
W
e
can
sumarize
Theorm
the
1
l
ling
Ther
curve
e
.
previous
exist
result
a
This
ge
o
in
desic
the
fol
laminto
laminto
wing
has
the
fol
on
Theorm:
the
lowing
pr
disk,
op
aso
ciate
d
to
Hilb
ert's
mo
die
d
sp
ac
e
sp
ac
e
ertis:
M
1.
The
end
l
p
ling
oints
of
curve
e
ach
elmnt
of
ar
e
map
e
d
to
the
same
p
oint
in
the
squar
e
by
the
.
M
2.
is
invart
under
the
gr
oup
of
symetri
of
the
squar
e,
the
dihe
dr
al
gr
oup
D
.
4
3.
The
laminto
F
has
=
2
a
tr
ansver
me
asur
e
and
ther
e
is
a
c
ontius
map
F
:
!
,
so
tha
.
4.
F
or
any
tr
ansver
ar
dimenso
is
1
=
2
.
c
A
to
nd,
,
the
ther
e
is
image
a
of
limt
set
this
on
limt
the
set
b
under
oundary
of
is
a
str
the
disk,
aight
whose
line
b
Hausdor
etw
en
the
p
oints,
M
whic
ar
e
the
images
under
of
the
end
p
oints
of
the
ge
o
desic,
joine
d
by
the
tr
ansver
ar
c.
M
4
Another
v
ing
In
section,
the
h
in
v
w
pa
whic
of
the
Hilb
ert's
curv
e
and
the
coresp
ond-
laminto
this
in
ersion
er.
has
t
wil
a
the
arin
e
In
sho
w
simlar
same
pro
under
a
w
the
a
v
y
ariton
of
as
b
ertis
efor,
as
diheral
Hilb
w
the
group
e
ert's
mo
a
previous
D
die
constru
one.
.
But
it
space
geo
Ho
is
in
v
ling
desic
w
arin
ev
t
curv
e
laminto
er
studie
aso
it
has
under
few
er
rotain
b
previously
ciated
to
this
symetri.
y
curv
It
1
=
e,
is
not
4.
4
^
W
e
consider
the
IFS,
f
^
h
;
^
h
;
0
^
h
;
1
h
g
2
on
the
in
terv
al:
3
8
t
1
15
+
if
0
t
<
<
4
64
16
^
h
(
t
)
=
0
:
t
15
15
if
4
^
64
h
(
t
)
=
h
(
k
)
+
k
=
=
4
for
k
=
1
;
2,
or
3.
Let
f
^
H
;
1
H
,
^
H
=
3
H
,
1
.
But
^
H
;
0
^
0
previously
t
0
^
H
<
;
^
^
and
t
16
H
=
1
w
e
2
mo
^
H
;
1
H
g
2
b
e
the
IFS
on
R
,
den
b
3
^
die
H
and
H
2
=
H
.
the
3
orde
Her
H
is
in
0
whic
the
map
of
h
k
the
8
square
are
visted.
the
IFS
on
the
square
studie
y:
1
2
3
Figure
Figure
9:
8:
The
geo
Other
desic
v
0
ersion
of
laminto
the
aso
Hilb
ciated
ert's
to
mo
the
die
curv
other
v
e.
ersion
of
Hilb
ert's
curv
e.
^
The
space
ling
curv
e
:
I
!
R
is
den
in
the
same
w
a
y
as
b
efor:
M
^
^
(
t
)
=
lim
^
H
^
H
M
a
^
wher
t
=
lim
^
h
n
a
Up
to
the
re-lab
curv
e
t
should
The
is
wn
plane
2.
suc
hap
ens
h
in
tha
the
e
at
This
map
is
the
con
cen
tin
is
a
four
uos
;
surjectiv
of
the
es
the
e
IFS
on
R
square,
equal
are
and
(
the
t
)
=
(1
+
i
)
=
curv
in
is
fol
clo
c
e
ed
b
y
w
this
or
re-lab
an
its
el
o
lamintos
the
vist
(
e
t
ecaus
p
t
ti
clo
c
times
the
in
previous
k-wise)
orde,
indces
of
op
ling
four
in
the
diagonly
space
oin
studie
k-wise
w
w
b
osite
as
the
subqare,
IFS
8
a
maps
simlar
w
and
the
a
y
one
coresp
can
obtain
ondig
a
lamintos
space
ling
curv
ha
v
9
e
e
8-side
on
to
the
h
yp
cub
erb
e
olic
in
p
R
olygns.
.
Her
the
on
as
8.
3
In
of
2.
j
thes
mainly
laminto
when
h
square,
is
The
visted
whic
the
This
and
laminto
Figure
on
.
sub-qare.
second
sub-qare
in
of
curv
tre
to
the
wn
maps
sub-qare
obtain
sho
the
ling
in
ther
orde
).
space
the
W
I
of
ort
when
gure
)
M
dive
obtained
in
(
indces
sup
sub
R
n
n
nish
is
(
a
^
h
the
h
and
square
section
the
whic
H
a
of
o
star
toal.
sho
w
2
eling
only
a
1
are
2
!1
^
h
!1
a
1
n
IFS
consit
Figure
5
10:
Geo
Space
In
section
desic
pro
ofs
w
wil
w
inscrb
the
Let
ed
case
us
n
sup
ev
ose
they
en
unit
tha
o
H
g
,
k
k
the
=0
;:n
n
IFS
i.e
wher
(
in
G
)
and
the
H
(
j
imp
z
ev
)
-gon
and
map
simlar
the
to
ha
in
those
ving
z
=
1
to
in
as
the
curv
a
the
one
e
regula
and
n
previous
of
Hexagon.
lamintos
ling
al
en
on
desic
space
terv
giv
es
geo
of
tha
and
curv
its
aso
-gon.
W
section.
its
v
ciated
e
omit
Let
ertics.
Ther
the
G
are
b
e
the
dierncs
d.
is
(
ling
en.
=
W
2
k
e
wil
consider
the
fol
wing
IFS
on
the
regula
n
-gon:
=n
e
(
z
+
1)
=
2.
The
atrco
of
this
IFS
is
the
n
-gon,
G
.
Unlik
e
k
studie
H
H
1
space
construi
disk
and
n
es
are
i
f
the
curv
the
to
previous
ling
in
ciated
on
the
space
since
-gon
en
es
genraliz
computains
n
et
e
to
and
aso
curv
laminto
regula
b
lamintos
ling
this
geo
desic
previous
G
)
section
in
tersc
this
in
a
set
IFS,
of
p
if
ositv
e
n
6=
Leb
4,
do
esgu
es
not
satify
measur,
the
for
op
some
j
en
6=
set
k
.
condit,
But
it
has
the
k
ortan
t
pro
ert
y:
the
in
terscion
of
al
H
(
G
)
is
the
orign,
the
geomtrical
cen
tre
of
the
n
-gon.
k
Let
f
h
g
b
k
k
=0
;:
;n
e
an
IFS
on
the
in
terv
al
I
,
wher
1
8
t
n
1
n
+
if
0
t
+1
<
<
2
n
h
(
t
)
2
n
2
n
=
0
:
t
n
1
1
n
+1
if
t
<
1
;
2
n
and
h
(
t
)
=
h
(
k
ling
t
)
+
k
=n
for
k
=
1
;
2
:
:
:
;
n
n
1.
2
In
a
n
simlar
w
a
y
,
as
b
efor,
w
e
den
the
space
0
curv
e
as:
(
t
)
=
lim
H
m
G
(
n
H
!1
a
is
w
el
den,
extrmis
con
of
h
(
I
H
(
a
G
),
wher
t
=
lim
h
a
m
h
!1
a
h
(
a
I
).
This
a
)
1
map
n
tin
),
is
uos
and
the
2
m
surjectiv
orign
z
e.
=
0,
the
F
cen
1
urthemo
the
ter
of
the
n
images
of
2
the
p
m
oin
ts
k
=n
,
i.e
the
-gon.
k
As
in
This
the
previous
section,
laminto
is
in
v
w
arin
t
e
can
under
aso
the
ciate
a
diheral
geo
desic
group
laminto
D
,
to
i.e
it
has
this
the
space
ling
same
curv
symetri
e.
as
the
n
n
-gon.
W
pro
e
can
ertis
stil
pro
exp
v
alid.
the
e
in
a
in
This
when
v
ounde
is
limt
due
set
simlar
the
is
a
1.
to
C
w
Theorm
fact
y
as
in
the
previous
Althoug
section
the
tha
w
construed.
e
So
IFS
on
elimnat
w
the
e
end
up
tha
the
n
has
cylinders
with
this
-gon
tha
sub-gon
laminto
o
v
Prop
cause
tha
has
erlaps,
o
do
v
not
erlaps
v
the
3.6
on
ha
al
ositn
e
o
v
the
is
n
erlaps
on
-gon,
their
Æ
in
terios.
W
If
w
e
e
sumarize
the
re-lab
el
the
pro
indces
ertis
of
of
the
IFS
this
laminto
on
the
in
plane
so
Theorm
tha
2.
H
(
G
)
are
arnged
an
ti
clo
c
k-wise
or
k
clo
c
k-wise
w
e
get
the
same
laminto.
Ho
w
ev
er
if
w
e
arnge
the
sub-gon
H
(
G
)
suc
h
tha
ther
at
In
is
k
least
one
this
sub-gon
case
w
whic
e
ha
v
e
to
h
is
fol
den
w
a
new
ed
b
y
IFS
its
diagonly
on
op
the
in
terv
osite,
al.
F
w
or
e
n
-gon,
whic
h
has
the
desir
pro
ert
y:
f
a
w
^
the
obtain
instace
e
g
,
k
=0
;:
;n
wher
H
=
1
H
the
indces:
8
0
if
j
if
n=
if
1
=
0
<
n
j
r
(
j
)
+
1
2
j
n
1
=
2
:
n
fol
and
k
r
on
t
the
laminto.
wing
IFS
on
^
H
k
functio
diern
use
j
10
j
<
n=
2
:
(
j
)
r
is
the
re-lab
el
Figure
1:
Geo
desic
laminto
aso
ciated
to
a
space
ling
curv
e
on
the
triangle.
^
W
e
consider
the
IFS,
f
h
g
on
k
k
=0
;:
;n
the
in
terv
al,
wher
1
8
t
n
2
1
+
n
+
if
0
t
+1
<
<
2
n
2
n
2
n
2
n
^
h
(
t
)
=
0
:
t
3
n
+1
if
t
<
1
;
2
n
^
and
2
n
2
n
^
h
(
t
)
=
^
h
(
k
t
)
+
k
=n
for
k
=
1
;
:
:
:
;
n
1.
In
a
simlar
w
a
y
w
e
obtain
a
space
ling
curv
e
and
0
G
(
n
)
^
a
geo
desic
aso
laminto
ciated
(
to
.
G
the
diheral
(
n
n
Ho
w
).
ev
This
er
geo
the
desic
laminto
has
symetri
are
the
diern
t.
same
pro
This
ertis
as
laminto
is
the
not
laminto
in
v
arin
t
under
)
group
D
.
It
is
in
v
arin
t
under
rotains
b
y
1
=n
.
In
sumary
w
e
get
the
fol
wing
n
Theorm:
^
Theorm
2
L
et
G
(
n
)
b
e
the
r
e
gular
n
-gon,
for
n
evn.
L
et
,
G
(
n
b
)
G
(
n
e
the
sp
ac
e
l
ling
curves
)
^
fr
om
the
interval
to
the
n
-gon,
den
d
ab
ove.
Then,
ther
e
ar
e
ge
o
desic
lamintos
(
n
)
and
(
n
)
on
^
the
disk,
aso
ciate
d
to
and
G
(
n
r
)
G
(
n
esp
e
ctively.
Thes
lamintos
have
the
fol
lowing
pr
op
ertis:
)
^
1.
The
end
p
oints
of
the
elmnts
of
(
n
)
(
(
n
)
)
ar
e
map
e
d
to
the
same
p
oint
in
the
n
-gon
by
^
the
sp
ac
e
l
ling
curve
(
G
2.
(
n
)
is
invart
(
n
under
).
)
G
the
(
gr
n
)
oup
of
symetri
of
the
n
-gon,
the
dihe
dr
al
gr
oup
D
.
A
nd
n
^
3.
(
F
n
)
or
is
invart
any
under
tr
ansver
the
ar
dimenso
is
s
=
c
r
to
log
(
2
=
n
otain
by
)
log
,
n
ther
e
.
A
1
is
a
=n
nd,
.
limt
set
the
on
image
the
of
b
this
oundary
of
limt
set
the
disk,
under
whose
Hausdor
is
a
str
aight
line
0
G
b
etw
en
the
p
oints,
whic
ar
e
the
images
under
of
G
(
n
the
end
p
oints
of
(
n
)
the
ge
o
desic,
joine
d
)
^
by
4.
the
tr
Each
ansver
ar
laminto
c.
Simlary
has
a
for
tr
(
ansver
me
n
)
asur
.
e
1
and
ther
e
is
a
c
ontius
map
F
:
(
n
)
!
(
n
)
s
0
^
(or
F
:
^
(
n
)
!
(
n
)
),
so
tha
F
=
n
.
If
n
is
o
d,
w
e
consider
the
fol
wing
IFS
on
the
plane:
f
H
g
,
k
i
e
2
k
k
=0
;:
;n
wher
H
1
(
z
)
k
=n
(
Lz
+
1
L
)
and
L
=
1
=
(1
+
cos(
=n
).
1
The
atrco
of
this
IFS
is
the
n
-gon.
Her
some
=
images
of
tion
of
G
under
al
the
H
(
G
)
maps
is
of
the
the
orign,
IFS,
the
o
cen
v
erlap
tre
in
of
the
a
n
set
of
-gon
G
p
.
ositv
e
The
Leb
IFS
esgu
on
measur.
the
in
The
terv
al
is:
f
in
h
tersc-
g
k
,
k
k
=0
;:
;n
1
wher
8
2
t
1
1
+
n
+
if
0
t
n
1
<
<
2
n
h
(
t
)
3
n
2
n
n
=
0
:
2
t
1
1
+
1
and
h
(
t
)
=
h
(
k
same
t
)
+
k
=n
for
n
=
;
:
:
;
1
n
n
ertis.
rotain
b
And
y
1
Theorm
=n
.
3
L
w
e
obtain
a
Sumarizng
w
et
G
(
n
)
b
e
the
geo
1.
r
e
e
desic
W
e
den,
as
get
laminto
the
on
b
efor,
gular
fol
n
wing
-gon,
the
to
ciate
d
the
to
n
The
end
l
(
p
ling
-gon,
.
G
1.
n
(
3.
F
;
space
ling
whose
curv
e
This
)
of
or
e
with
the
n
elmnt
of
(
n
en
b
y
o
d.
ther
L
fol
e
is
et
lowing
a
pr
ge
b
op
o
(
n
e
the
sp
ac
e
l
ling
curve
fr
om
)
desic
lamintos
(
n
)
on
the
disk,
ertis:
c
tr
dimenso
(
n
)
ar
e
map
e
d
to
the
same
p
oint
in
the
n
-gon
by
the
sp
ac
e
)
under
ar
Then,
the
giv
Theorm:
for
ove.
are
.
invart
any
ab
has
ach
is
d
laminto
oints
curve
n
den
symetri
)
G
2.
1
a
disk,
G
interval
aso
<
n
0
pro
the
t
2
n
:
n
if
3
n
0
n
+
2
n
the
ansver
is
r
to
s
=
log
(
2
=
n
otain
log
by
)
n
,
ther
e
1
is
=n
a
.
limt
set
on
the
b
oundary
of
the
disk,
whose
Hausdor
.
0
4.
Each
laminto
has
a
1
tr
ansver
me
asur
e
and
ther
e
is
a
c
ontius
map
F
:
(
n
)
!
(
n
)
s
0
so
tha
F
=
n
.
Ac
IFS
kno
wledgmn
on
ts:
the
n
previous
v
-gon.
The
ersion
The
author
author
of
w
this
is
pa
also
ould
lik
grateful
e
to
to
thank
the
An
ref
thon
y
whose
Manig
for
comen
ts
p
help
oin
ting
ed
out
to
the
impro
v
e
the
er.
Refrncs
[1]
P
.
le
Arnoux,
Un
[2]
M.F
[3]
D.
tore,
Bul
exmpl
l.
de
So
Barnsley
c.
,
F
Math.
r
semi-conjuga
F
actls
r
en
anc
e
,
16
evrywh
(198),
e
,
tre
un
ec
hange
d'in
terv
ales
et
une
rotain
sur
489-50.
Second
Editon,
Academi
Pres
Profesinal,
Bostn,
193.

Hilb
ert,
38
Ub
[4]
(189),
K.
Kelr,
in
er
die
Invarit
Leb
factors,
esgu,
L
Vilars,
1904,
G.
P
P
eano,
einr
A
e
e
Springe,
cons
Line
bhand
Julia
1732,
H.
[6]
Abildung
Gesamlt
Mathemics,
[5]
steig
459-60.
l'Int
ein
V
quivalenc
es
Berlin,
sur
auf
lunge
ol
and
I
Fl
I
the
I,
ac
henst
1-2,
uc

k,
Mathemisc
Springe-V
(abstr
A
erlag,
act)
Berlin,
Mandelbr
ot
nale
set
1935.
,
Lectur
Notes
egr
20.
ation
et
la
R
e
cher
che
des
F
onctis
Primtves
,
Gauthiers-
Sur
aris.
une
courb
e
qui
remplit
toue
une
aire
plane,
Mathemisc
A
nale
36
(1890),
157-60.
[7]
G.
P
eano,
1908,
La
[8]
F
H.
Bo
Sagn,
40
[9]
a
di
ca,
T
Apro
(192),
ximatng
P
eano
Nel
urin,
pgs.
P
`F
orm
ulario
Mathemico',
F
ormulai
Mathemic
o
,
v
239-40.
olygns
for
the
Sierp
nski-Kop
Curv
e,
Bul
l.
A
c
ad.
Sc.
Polnaise
19-2.
Sagn,
aise
H.
,
Curv
rates
No
40
,
(192),
wher
diern
tiabl
y
of
the
Sierp
nski'
217-0.
12
space-ling
curv
e,
Bul
l.
A
c
ad.
Sc.
Poln-
ol
5,
,
[10]
H.Sagn,
[1]
I.
Sp
J.
Sc
ho
ac
en
b
e-Fil
ling
erg,
On
curves
,
the
P
Springe-V
eano
erlag,
curv
e
of
New
Leb
Y
esgu,
ork,
Bul
194.
l.
of
the
A
mer.
Math.
So
c.
,
4
,
(1938),
519.
[12]
W.
Sierp
nski,
Sci.
[13]
de
Cr
V.F
t,
en
and
V.F
t,
nouv
eri
el
A,
lamintos
Arnoux
tin
,
qui
remplit
geomtric
toue
une
realiztons
is
259
,
lamintos
Math.
ue
aire
plane,
Bul
l.
A
c
ad.
c.
Simon
Pisot
subtion,
Er
go
dic
H
older
con
tin
uos,
Journal
of
Mathemic
al
A
nal-
357-6.
as
So
of
1253-6.
semi-conjugay
desic
Belg.
con
as
20
(201),
Geo
l.
e
462-78.
(20),
The
t,
Bul
courb
(192),
desic
ations
en
in
Geo
Aplic
Sirv
ear
S
System
Sirv
ysi
ap
,
Dynam.
V.F
[15]
une
ovie
en
ory
[14]
Sur
ac
Sirv
The
geomtric
realiztons
Stevin
13
.
of
Arnoux-Razy
sequnc.
T
o