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DYNAMICAL SYSTEMS TRANSFER OPERATORS

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DYNAMICAL SYSTEMS, TRANSFER OPERATORS and FUNCTIONAL ANALYSIS Brigitte Vallee, Laboratoire GREYC (CNRS et Universite de Caen) Seminaire CALIN, LIPN, 5 octobre 2010

  • euclidean algorithm

  • probabilistic analysis

  • seminaire calin

  • analytical properties

  • generating function

  • transfer operator

  • universite caen

  • dynamical analysis


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Informations

Publié par
Publié le 01 octobre 2010
Nombre de lectures 89

DYNAMICALSYSTEMS,TRANSFEROPERATORS

andFUNCTIONALANALYSIS

Brigitte
Valle´e
,LaboratoireGREYC

(CNRSetUniversite´deCaen)

Se´minaireCALIN,LIPN,5octobre2010

mhtiroglAnaedilcuEehtfosisylanacitsilibaborP⇓noitcnufgnitarenegehtfoseitreporplacitylanA⇓rotareporefsnartehtfoesrevnI-isauQehtfoseitreporplacitylanA⇓rotareporefsnartehtfoseitreporplartcepS⇓sehcnarbehtfoseitreporpcirtemoeG⇓noisividehtfoseitreporpcitemhtirA⇓mhDynamical

tanalysis

ifo

ra

oEuclidean

gAlgorithm
.

lAnaedilcuEA
edilcuEehtfosisylanacitsilibaborP⇓noitcnufgnitarenegehtfoseitreporplacitylanA⇓rotareporefsnartehtfoesrevnI-isauQehtfoseitreporplacitylanA⇓rotareporefsnartehtfoseitreporplartcepS⇓sehcnarbehtfoseitreporpcirtemoeGAEuclideanAlgorithm

Arithmetic

properties

fo

eht

division

Dynamical
analysis
ofaEuclidean
Algorithm
.

mhtiroglAna
mhtiroglAnaedilcuEehtfosisylanacitsilibaborP⇓noitcnufgnitarenegehtfoseitreporplacitylanA⇓transferoperator

Analytical
propertiesofthe
Quasi-Inverse
ofthe

Spectral
propertiesofthe
transferoperator

Geometric
propertiesofthe
branches

AEuclideanAlgorithm

Arithmetic
propertiesofthe
division

Dynamical
analysis
ofaEuclidean
Algorithm
.

Dynamical
analysis
ofaEuclidean
Algorithm
.

AEuclideanAlgorithm

Arithmetic
propertiesofthe
division

Geometric
propertiesofthe
branches

Spectral
propertiesofthe
transferoperator

Analytical
propertiesofthe
Quasi-Inverse
ofthe

transferoperator

Analytical
propertiesofthe
generatingfunction

Probabilisticanalysis
oftheEuclidean
Algorithm

input
(
u,v
)
,itcomputesthe
gcd
of
u
and
v
,

ehtnO

together

htiw

eht

deunitnoC

rFnoitca

noisnapxE

fo

.v/u

ehT

(standard)EuclidAlgorithm:thegrandfatherofallthealgorithms.

,pm1+...1+2m1+1m1=vu:vufoEFC.htpedehtsip.stigidehteras’imeht,vdnaufodcgehtsipu0=1+pu0+pupm=1−pu1−pu<pu<0pu+1−pu1−pm=2−pu+...=...2u<3u<03u+2u2m=1u1u<2u<02u+1u1m=0u1u≥0u;u=:1u;v=:0u
The(standard)EuclidAlgorithm:thegrandfatherofallthealgorithms.

Ontheinput
(
u,v
)
,itcomputesthe
gcd
of
u
and
v
,

togetherwiththe
ContinuedFractionExpansion
of
u/v

.v/ufonoisnapxEnoitcarFdeunitnoC

u=:u;v=:uu;100

u
0
=
m
1
u
1
+
u
2


u
1
=
m
2
u
2
+
u
3
...
=
...
+

u
p

2
=
m
p

1
u
p

1
+
u
p
u
p

1
=
m
p
u
p
+0

u
p
isthe

≥u1

0
<u
2
<u
1
0
<u
3
<u
2

0
<u
p
<u
p

1

u
p
+1
=0

.stigidehteras’meht,vdnaufodcg.htpedehtsipi

FCEfouv:uv=m1+m21+.1..1+1mp,
The(standard)EuclidAlgorithm:thegrandfatherofallthealgorithms.

Ontheinput
(
u,v
)
,itcomputesthe
gcd
of
u
and
v
,

togetherwiththe
ContinuedFractionExpansion
of
u/v

unitnoC.v/ufonoisnapxEnoitcarFde

u
0
:=
v
;
u
1
:=
u
;
u
0

u
1

u
0
=
m
1
u
1
+
u
2


u
1
=
m
2
u
2
+
u
3
...
=
...
+

u
p

2
=
m
p

1
u
p

1
+
u
p
u
p

1
=
m
p
u
p
+0

0
<u
2
<u
1
0
<u
3
<u
2

0
<u
p
<u
p

1

u
p
+1
=0

u
p
isthe
gcd
of
u
and
v
,the
m
i
’sarethe
digits
.
p
isthe
depth
.

uCFEof:
v

1u,=1v+m11+m21...+mp

.mhtiroglaehtfonoisnetxesuounitnocasimetsyslacimanydehT.0sehcaertahtyrotcejarta=)T,]1,0[(metsySlacimanyDehtfoyrotcejartlanoitarA=)0,...,)x(2T,)x(T,x(mhtiroglAnaedilcuEehtfonoitucexenA0=)0(T,0=6xrofx1−x1=:)x(T,]1,0[→−]1,0[:Terehw,)ix(T=1+ixroix1−ix1=1+ixsanettirwnehtsi1+iu+iuim=1−iunoisividehT.1−iuiu=:ixlanoitarehtyb)1−iu,iu(riapregetniehtecalpeRTheunderlyingEuclideandynamicalsystem(I).

(
u
1
,u
0
)
is:

ThetraceoftheexecutionoftheEuclidAlgorithmon

(
u
1
,u
0
)

(
u
2
,u
1
)

(
u
3
,u
2
)

...

(
u
p

1
,u
p
)

(
u
p
+1
,u
p
)=

(0
,u
p
)

.mhtiroglaehtfonoisnetxesuounitnocasimetsyslacimanydehT.0sehcaertahtyrotcejarta=)T,]1,0[(metsySlacimanyDehtfoyrotcejartlanoitarA=)0,...,)TheunderlyingEuclideandynamicalsystem(I).

xfor
x
6
=0
,T
(0)=0

(11T
:[0
,
1]
−→
[0
,
1]
,T
(
x
):=
x

x

211x
i
+1
=
x

x
or
x
i
+1
=
T
(
x
i
)
,
where
ii

TThedivision
u
i

1
=
m
i
u
i
+
u
i
+1
isthenwrittenas

,uReplacetheintegerpair
(
u
i
,u
i

1
)
bytherational
x
i
:=
i
.
u1−i

)ThetraceoftheexecutionoftheEuclidAlgorithmon
(
u
1
,u
0
)
is:

x(
u
1
,u
0
)

(
u
2
,u
1
)

(
u
3
,u
2
)

...

(
u
p

1
,u
p
)

(
u
p
+1
,u
p
)=(0
,u
p
)

(T,x(mhtiroglAnaedilcuEehtfonoitucexenA