DYNAMICAL SYSTEMS TRANSFER OPERATORS
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57
pages
Documents
2010
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
- euclidean algorithm
- probabilistic analysis
- seminaire calin
- analytical properties
- generating function
- transfer operator
- universite caen
- dynamical analysis
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,vdnaufodcgehtsipu0=1+pu0+pupm=1−pu1−pu<pu<0pu+1−pu1−pm=2−pu+...=...2u<3u<03u+2u2m=1u1u<2u<02u+1u1m=0u1u≥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
