On the Hausdorff Dimension of the Mather Quotient

OntheHausdorffDimensionoftheMather
Quotient

AlbertFathi,
∗
AlessioFigalli,
†
LudovicRifford
‡
8November,2007

Abstract
Underappropriateassumptionsonthedimensionoftheambientman-
ifoldandtheregularityoftheHamiltonian,weshowthattheMatherquo-
tientissmallintermofHausdorffdimension.Then,wepresentapplications
indynamics.

1Introduction
Let
M
beasmoothmanifoldwithoutboundary.Wedenoteby
TM
thetangent
bundleandby
π
:
TM
→
M
thecanonicalprojection.Apointin
TM
willbe
denotedby(
x,v
)with
x
∈
M
and
v
∈
T
x
M
=
π
−
1
(
x
).Inthesamewayapoint
ofthecotangentbundle
T
∗
M
willbedenotedby(
x,p
)with
x
∈
M
and
p
∈
T
x
∗
M
alinearformonthevectorspace
T
x
M
.Wewillsupposethat
g
isacomplete
Riemannianmetricon
M
.For
v
∈
T
x
M
,thenorm
k
v
k
x
is
g
x
(
v,v
)
1
/
2
.Wewill
denoteby
k∙k
x
thedualnormon
T
∗
M
.Moreover,foreverypair
x,y
∈
M
,
d
(
x,y
)
willdenotetheRiemanniandistancefrom
x
to
y
.
Wewillassumeinthewholepaperthat
H
:
T
∗
M
→
R
isanHamiltonianof
class
C
k,α
,with
k
≥
2
,α
∈
[0
,
1],whichsatisfiesthethreefollowingconditions:
(H1)C
2
-strictconvexity:
∀
(
x,p
)
∈
T
∗
M
,thesecondderivativealongthefibers
∂
2
H/∂p
2
(
x,p
)isstrictlypositivedefinite;
(H2)
uniformsuperlinearity:
forevery
K
≥
0thereexistsafiniteconstant
C
(
K
)suchthat
∀
(
x,p
)
∈
T
∗
M,H
(
x,p
)
≥
K
k
p
k
x
+
C
(
K
);
∗
UMPA,ENSLyon,46Alle´ed’Italie,69007Lyon,France.
e-mail:albert.fathi@umpa.ens-
lyon.fr
†
Universite´deNice-SophiaAntipolis,ParcValrose,06100Nice,France.
e-mail:fi-
galli@unice.fr
‡
Universite´deNice-SophiaAntipolis,ParcValrose,06100Nice,France.
e-mail:rif-
ford@unice.fr

1

(H3)
uniformboundednessinthefibers:
forevery
R
≥
0,wehave
sup
{
H
(
x,p
)
|k
p
k
x
≤
R
}
<
+
∞
.
M∈xBytheWeakKAMTheoremweknowthat,undertheaboveconditions,there
is
c
(
H
)
∈
R
suchthattheHamilton-Jacobiequation
H
(
x,d
x
u
)=
c
(HJ
c
)
admitsaglobalviscositysolution
u
:
M
→
R
for
c
=
c
(
H
)anddoesnotadmit
suchsolutionfor
c<c
(
H
),see[22,9,6,11,15].Infact,for
c<c
(
H
),the
Hamilton-Jacobiequationdoesnotadmitanyviscositysubsolution(forthetheory
ofviscositysolutions,wereferthereadertothemonographs[1,2,11]).Moreover,
if
M
isassumedtobecompact,then
c
(
H
)istheonlyvalueof
c
forwhichthe
Hamilton-Jacobiequationaboveadmitsaviscositysolution.Theconstant
c
(
H
)is
calledthe
criticalvalue
,orthe
Man˜e´criticalvalue
of
H
.Inthesequel,aviscosity
solution
u
:
M
→
R
of
H
(
x,d
x
u
)=
c
(
H
)willbecalleda
criticalviscositysolution
ora
weakKAMsolution
,whileaviscositysubsolution
u
of
H
(
x,d
x
u
)=
c
(
H
)will
becalleda
criticalviscositysubsolution
(or
criticalsubsolution
if
u
isatleastC
1
).
TheLagrangian
L
:
TM
→
R
associatedtotheHamiltonian
H
isdefinedby
∀
(
x,v
)
∈
TM,L
(
x,v
)=
p
∈
m
T
a
∗
x
M
{
p
(
v
)
−
H
(
x,p
)
}
.
xSince
H
isofclassC
k
,with
k
≥
2,andsatisfiesthethreeconditions(H1)-(H3),it
iswell-known(seeforinstance[11]or[15,Lemma2.1]))that
L
isfiniteeverywhere
ofclassC
k
,andisaTonelliLagrangian,i.e.satisfiestheanalogousofconditions
(H1)-(H3).Moreover,theHamiltonian
H
canberecoveredfrom
L
by
∀
(
x,p
)
∈
T
x
∗
M,H
(
x,p
)=max
{
p
(
v
)
−
L
(
x,v
)
}
.
MT∈vxThereforethefollowinginequalityisalwayssatisfied
p
(
v
)
≤
L
(
x,v
)+
H
(
x,p
)
.
ThisinequalityiscalledtheFenchelinequality.Moreover,duetothestrictcon-
vexityof
L
,wehaveequalityintheFenchelinequalityifandonlyif
(
x,p
)=
L
(
x,v
)
,
where
L
:
TM
→
T
∗
M
denotestheLegendretransformdefinedas
L∂L
(
x,v
)=
x,
(
x,v
)
.
v∂Underourassumption
L
isadiffeomorphismofclassatleast
C
1
.Wewilldenote
by
φ
tL
theEuler-Lagrangeflowof
L
,andby
X
L
thevectorfieldon
TM
that
2

generatestheflow
φ
tL
.Ifwedenoteby
φ
tH
theHamiltonianflowof
H
on
T
∗
M
,
thenasiswell-known,seeforexample[11],thisflow
φ
tH
isconjugateto
φ
tL
by
theLegendretransform
L
.
AsdonebyMatherin[26],itisconvenienttointroducefor
t>
0fixed,the
function
h
t
:
M
×
M
→
R
definedby
tZ∀
x,y
∈
M,h
t
(
x,y
)=inf
L
(
γ
(
s
)
,γ
˙(
s
))
ds,
0wheretheinfimumistakenoveralltheabsolutelycontinuouspaths
γ
:[0
,t
]
→
M
with
γ
(0)=
x
and
γ
(
t
)=
y
.The
Peierlsbarrier
isthefunction
h
:
M
×
M
→
R
definedby
h
(
x,y
)=li
t
m
→
i
∞
nf
{
h
t
(
x,y
)+
c
(
H
)
t
}
.
Itisclearthatthisfunctionsatisfies
∀
x,y,z
∈
M,h
(
x,z
)
≤
h
(
x,y
)+
h
t
(
y,z
)+
c
(
H
)
t
h
(
x,z
)
≤
h
t
(
x,y
)+
c
(
H
)
t
+
h
(
y,z
)
,
andthereforeitalsosatisfiesthetriangleinequality
∀
x,y,z
∈
M,h
(
x,z
)
≤
h
(
x,y
)+
h
(
y,z
)
.
Moreover,givenaweakKAMsolution
u
,wehave
∀
x,y
∈
M,u
(
y
)
−
u
(
x
)
≤
h
(
x,y
)
.
Inparticular,wehave
h>
−∞
everywhere.Itfollows,fromthetriangleinequal-
ity,thatthefunction
h
iseitheridentically+
∞
oritisfiniteeverywhere.If
M
iscompact,
h
isfiniteeverywhere.Inaddition,if
h
isfinite,thenforeach
x
∈
M
thefunction
h
x
(
∙
)=
h
(
x,
∙
)isacriticalviscositysolution(see[11]or[16]).The
projectedAubryset
A
isdefinedby
A
=
{
x
∈
M
|
h
(
x,x
)=0
}
.
FollowingMather,see[26,page1370],wesymmetrize
h
todefinethefunction
δ
M
:
M
×
M
→
R
by
∀
x,y
∈
M,δ
M
(
x,y
)=
h
(
x,y
)+
h
(
y,x
)
.
Since
h
satisfiesthetriangleinequalityand
h
(
x,x
)
≥
0everywhere,thefunc-
tion
δ
M
issymmetric,everywherenonnegativeandsatisfiesthetriangleinequality.
Therestriction
δ
M
:
A×A→
R
isagenuinesemi-distanceontheprojectedAubry
set.Wewillcallthisfunction
δ
M
the
Mathersemi-distance
(evenwhenwecon-
sideriton
M
ratherthanon
A
).Wedefinethe
Matherquotient
(
A
M
,δ
M