Constru tion of urious minimal uniquely ergodi

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Niveau: Secondaire, Lycée, Terminale
ar X iv :m at h. D S/ 06 05 43 8 v1 16 M ay 2 00 6 Constru tion of urious minimal uniquely ergodi homeomorphisms on manifolds: the Denjoy-Rees te hnique F. Beguin y , S. Crovisier z and F. Le Roux x 17th May 2006 Abstra t In [23?, Mary Rees has onstru ted a minimal homeomorphism of the 2-torus with pos- itive topologi al entropy. This homeomorphism f is obtained by enri hing the dynami s of an irrational rotation R. We improve Rees onstru tion, allowing to start with any homeomorphism R instead of an irrational rotation and to ontrol pre isely the measurable dynami s of f . This yields in parti ular the following result: Any ompa t manifold of dimension d 2 whi h arries a minimal uniquely ergodi homeomorphism also arries a minimal uniquely ergodi homeomorphism with positive topologi al entropy. More generally, given some homeomorphism R of a ( ompa t) manifold and some home- omorphism h C of a Cantor set, we onstru t a homeomorphism f whi h \looks like R from the topologi al viewpoint and \looks like R h C from the measurable viewpoint.

  • topologi al

  • ergodi homeomorphisms

  • minimal homeomorphism

  • minimal uniquely

  • constru tion

  • ting examples

  • stri tly

  • tion pro

  • measurable dynami


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oratoire
i
lik
Construction
generally
of
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minimal
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uniquely

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homeomorphisms
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oin
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math

al
eguin
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some
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tor
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the
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and
b
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to
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h
Roux
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x
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ematiques,
Ma
Cedex,
y
Analyse,
2006
Univ.

x
In
P
[23
rance.

opy.
Mary
giv
Rees
R
has
manifold

omorphism
a
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itiv
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ematiques,
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.

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arXiv:math.DS/0605438 v1 16 May 2006.
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3
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.
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of
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:
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1
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2
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27
3
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5
.
6
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's:
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The
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of
;
homeomorphisms
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;
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28
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(

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6.1
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othesis
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sc
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35
Main
The

(
of
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h
k
yp
1
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B
k
1
k
;
1
2
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21
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other
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I
the
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of
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h
.
on
.
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.
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set
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K
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51
C
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36
B
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of
Hyp
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othesis
.
C
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5
.
and
.
some
.

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homeomorphisms
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y:
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C
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58
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top
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of
.
8
37
.
8.2
.
The
9.5
b
.
ered
.
map
.
h
.
.
51
.
een
.
ransitivit
.
addendum
.
B
.
.
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h
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C
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54
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a
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with
.
y
.
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.
38
.
8.3
.
Hyp
9.4
othesis
yp
C
and
6
.
and
.
some
.

.
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of
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.
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.
et
.
tor
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.
,
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of
.
53
.
otheses
.
and
38
.
8.4
.
Construction
.
of
.
the
.

.
(
.
n
.
k
Realisation
):
otheses
realisation
and
of
.
h
.
yp
.
otheses
.
C
.
3
.
;
I.3
4
addendum
.
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38
.
8.5
58
Construction
Some
of
I
the

H
.
k
.
's:
.
realisation
.
of
.
h
.
yp
.
otheses
I
C
t
1
of
;
in
2

;
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5
.
;
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6
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I.3
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ositiv
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en
.
the
.
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iii
.
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.
39
.
8.6
.
First
.
part
44
of
Realisation
the
h
pro
otheses
of
7
of
C
theorem
.
1.3,
.
and
.
minimal
.
homeomorphisms
.
with
.
p
.
ositiv
.
e
.
top
.
ological
.
en
45
trop
Pro
y
of
.
1.3
.
.
.
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50
.

.
I
.
of
.
b
.
w
.
Can
.
sets
.
I
.
T
.
y
41
minimalit
9
pro
Suppressing
of
the
1.4

I
outside
Hyp
the
f
Can
4
tor
f
set
4
K
.

.
C
.
42
.
9.1
.
The
.
w
.
aste
.
bins
.
(
.
P
.
i
.
)
.
i
.
2
54
N
I.2
and
of
the
yp
neigh
f
b
4
ourho
f
o
4
ds
.
(
.
V
.
k
.
)
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k
.
2
.
N
.
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I
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Pro
.
of
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1.4
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42
.
9.2
.
Hyp
.
otheses
.
C
.
7
.
;
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8
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.
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I
.
I
.
examples
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I
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I.1
.
y
.
ter-examples
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58
.
I
.
Dieren
.
w
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blo
.
an
.
v
.
t
43
.
9.3
.
Consequences
.
of
.
h
.
yp
.
otheses
.
C
.
7
59
;
I
8
Pseudo-rotations
.
p
.
e
.
ological
.
trop
.
on
.
2-sphere
.
.
.
.
.
60
.
.trop
1
\blo
In
the
tro
The

or
1.1
dimension
Denjo
example,
y-Rees
particular

to
hnique
homeomorphism
Tw
er
en
lo
t
Rees
y-v
has
e
what
y
Rees
ears
h
ago,
minimal
M.
manifolds.
Rees
ergo
has
2

athi
a

homeomorphism
a
of
y
the

torus
e
T
orbits
d
is
(
te
d
y

d
2)
will
whic
homeomorphisms
h
new
is
ositiv
minimal
y
and
ositiv
has
go
p

ositiv

e

top
top
ological
v
en
free
trop
of
y
no
(see
d
[23
just


The
h
existence
of
of
homeomorphism

trop
h
up
an
e
example
presen
is
a
surprising
the
for
general
sev
of
eral
an
reasons:
minimal
{
e

v
examples
new
of

minimal
\do
homeomorphisms
h
(irrational
as
rotations,

time
en
t
y
maps
as
of

horo
en

b
o
is
ws,
application

will
are
Theorem
also
of
t
a
ypical

examples
omorphism
of
entr
zero
ha
en
ery
trop
admitting
y

maps.
(see
{
said
a
fr

action
w
1
a
and
y
some
for
in
pro
ter-example.
ving

that
m
a

map
the
f
y;
has
get
p
p
ositiv
ological
e
,
top
blo
ological
set
en
p
trop
esgue
y
of
is
pap
to
describ
sho
setting
w
e
that
e
the
.
n

um
the
b
arious
er
ter-examples"
of
dimension,
p
of
erio
of

p
orbits
ological
of
.
p
w
erio
elop
d
hnique

ws
n
that
for
e
f

gro
m
ws
This
exp

onen
existence
tially
er
fast
with
when
top
n
y
!
p
1
realise
.

So,
homeomorphisms
in

man
with
y
top
situations,
y
\p
said
ositiv

e
if
top
and
ological
As
en
Denjo
trop
w
y"
v
is
wing
synon
A
ymous
act
of
d
\man

y
er
p
omorphism
erio
a


orbits".
p
But

a
A.
minimal
M.
homeomorphism
e
do
that
not
manifold
ha

v
lo
e
of
an

y

p
An
erio


b
orbit.
al
{
e
a
of
b

eautiful
oin
theorem
T
of
,
A.
to
Katok
w-up"
states
orbits,
that,
as
if
Denjo
f

is
Of
a
the
C
of
1+
is

uc
dieomorphism
more
of
and
a
than

one
surface
Denjo
S
for
with
to
p
a
ositiv
with
e
ositiv
top
top
ological
en
en
y
trop
one
y
to
,
w
then
a
there
of
exists
of
an
ositiv
f
Leb
-in
measure.
v
aim
arian
the
t
t

er
set
to

e

general
S
for

w
h

that
Denjoy-R
some
es
p

o
This
w
setting
er
as
of

f

j
v

\Denjo
is


in
to
y
a
and
full

shift
a
(see
homeomorphism
[18
T
,
with

ositiv
4.3]).
top
In
en
particular,
y
a
Moreo
C
er,
1+
e

dev
dieomorphism
a
of

a
whic

allo
surface
to
with
trol
p
the
ositiv
w
e
obtain
top
not
ological
tain
en
o
trop
uc
y


yields
b
results
e
h
minimal.
the
Bey
of
ond
uniquely
the
go
mere
homeomorphisms
existence
p
of
e
minimal
ological
homeomorphisms
trop
of
,
T
the
d
ossibilit
with
to
p
man
ositiv
measurable
e
systems
top
minimal
ological
on
en
1.2
trop
ergo
y
homeomorphisms
,
p
the
e

ological
hnique
trop
used
A
b
is
y
to
Rees
e
to
er



it
h
minimal
a
uniquely
homeomorphism

is
an
v
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