On the total disconnectedness of the quotient Aubry set
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT
AUBRY SET
ALFONSO SORRENTINO
Abstra
t. In this paper we show that the quotient Aubry set, asso
iated
to a su�
iently smooth me
hani
al or symmetri
al Lagrangian, is totally dis-

onne
ted (i.e., every 
onne
ted 
omponent 
onsists of a single point). This
result is optimal, in the sense of the regularity of the Lagrangian, as Mather's

ounterexamples in [19℄ show. Moreover, we dis
uss the relation between this
problem and a Morse-Sard type property for (di�eren
e of) 
riti
al subsolu-
tions of Hamilton-Ja
obi equations.
1. Introdu
tion.
In Mather's studies of the dynami
s of Lagrangian systems and the existen
e of
Arnold di�usion, it turns out that understanding 
ertain aspe
ts of the Aubry set
and, in parti
ular, what is 
alled the quotient Aubry set, may help in the 
onstru
-
tion of orbits with interesting behavior.
While in the 
ase of twist maps (see for instan
e [3, 12℄ and referen
es therein)
there is a detailed stru
ture theory for these sets, in more degrees of freedom quite
few is known. In parti
ular, it seems to be useful to know whether the quotient
Aubry set is �small� in some sense of dimension (e.g., vanishing topologi
al or box
dimension).
In [18℄ Mather showed that if the state spa
e has dimension ≤ 2 (in the non-
autonomous 
ase) or the Lagrangian is the kineti
 energy asso
iated to a Riemann-
ian metri
 and the state spa
e has dimension ≤ 3, then the quotient Aubry set is
totally dis
onne
ted, i.e., every 
onne
ted 
omponent 
onsists of a single point (in
a 
ompa
t metri
 spa
e this is equivalent to vanishing topologi
al dimension). In
the autonomous 
ase, with dimM ≤ 3, the same argument shows that this quotient
is totally dis
onne
ted as long as the Aubry set does not interse
t the zero se
tion
of TM (this is the 
ase when the 
ohomology 
lass is large enough in norm).
What happens in higher dimension? Unfortunately, this is generally not true. In
fa
t, Burago, Ivanov and Kleiner in [6℄ provided an example that does not satisfy
this property (they did not dis
uss it in their work, but it follows from the results
therein). More strikingly, Mather provided in [19℄ several examples of quotient
Aubry sets that are not only non-totally-dis
onne
ted, but even isometri
 to 
losed
intervals. All these examples 
ome from me
hani
al Lagrangians on TTd (i.e., the
sum of the kineti
 energy and a potential) with d ≥ 3. In parti
ular, for every ε > 0,
he provided a potential U ∈ C2d−3,1−ε(Td), whose asso
iated quotient Aubry set is
isometri
 to an interval. As the author himself noti
ed, it is not possible to improve
the di�erentiability of these examples, due to the 
onstru
tion 
arried out.
The main aim of this arti
le is to show that the 
ounterexamples provided by
Mather are optimal, in the sense that for more regular me
hani
al Lagrangians, the
asso
iated quotient Aubry sets - 
orresponding to the zero 
ohomology 
lass - are
totally dis
onne
ted.
In parti
ular, our result will also apply to slightly more general Lagrangians, satis-
fying 
ertain 
onditions on the zero se
tion; in this 
ase, we shall be able to show
http://arxiv.org/abs/0704.0129v1
2 ALFONSO SORRENTINO
that the quotient Aubry set, 
orresponding to a well spe
i�ed 
ohomology 
lass, is
totally dis
onne
ted.
We shall also outline a possible approa
h to generalize this result, pointing out how
it is related to a Morse-Sard type problem; from this and Sard's lemma, one 
an
easily re
over Mather's result in dimension d = 2 (autonomous 
ase).
It is important to point out, that most of this approa
h has been inspired by Albert
Fathi's talk [9℄, in whi
h he used this relation with Sard's lemma to show a simpler
way to 
onstru
t me
hani
al Lagrangians on TTN , whose quotient Aubry sets are
Lips
hitz equivalent to any given doubling metri
 spa
e or, equivalently, to any spa
e
with �nite Assouad dimension (see [13℄ for a similar 
onstru
tion). In this 
ase we
do not get a neat relation between their regularity and N , as in Mather's, but we

an only observe that N goes to in�nity as r in
reases. It would be interesting to
study in depth the relation between the dimension of the quotient Aubry set, the
regularity of the Lagrangian and the dimension of the state spa
e. Our result may
be seen as a �rst step in this dire
tion.
Post S
riptum. Just before submitting this paper, we learnt that analogous results
had been proven indipendently by Albert Fathi, Alessio Figalli and Ludovi
 Ri�ord,
using a similar approa
h (to be published).
Moreover, in �A generi
 property of families of Lagrangian systems� (to appear on
Annals of Mathemati
s), Patri
k Bernard and Gonzalo Contreras managed to show
that generi
ally, in Mañé's sense, there are at most 1+dimH1(M ;R) ergodi
 min-
imizing measures, for ea
h 
ohomology 
lass c ∈ H1(M ;R). As a 
orollary of this
striking result, one gets that generi
ally the quotient Aubry set is �nite for ea
h

ohomology 
lass and it 
onsists of at most 1 + dimH1(M ;R) elements.
2. The Aubry set and the quotient Aubry set.
Let M be a 
ompa
t and 
onne
ted smooth manifold without boundary. Denote
by TM its tangent bundle and T∗M the 
otangent one. A point of TM will be
denoted by (x, v), where x ∈ M and v ∈ TxM , and a point of T∗M by (x, p), where
p ∈ T∗xM is a linear form on the ve
tor spa
e TxM . Let us �x a Riemannian metri
g on it and denote with d the indu
ed metri
 on M ; let ‖ · ‖x be the norm indu
ed
by g on TxM ; we shall use the same notation for the norm indu
ed on T
De�nition. A fun
tion L : TM −→ R is 
alled a Tonelli Lagrangian if:
i) L ∈ C2(TM);
ii) L is stri
tly 
onvex in the �bers, i.e., the se
ond partial verti
al derivative
(x, v) is positive de�nite, as a quadrati
 form, for any (x, v) ∈ TM ;
iii) L is superlinear in ea
h �ber, i.e.,
‖v‖x→+∞
L(x, v)
(this 
ondition is independent of the 
hoi
e of the Riemannian metri
).
Given a Lagrangian, we 
an de�ne the asso
iated Hamiltonian, as a fun
tion on the

otangent bundle:
H : T∗M −→ R
(x, p) 7−→ sup
v∈TxM
{〈p, v〉x − L(x, v)}
where 〈 ·, · 〉x represents the 
anoni
al pairing between the tangent and 
otangent
spa
e.
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 3
If L is a Tonelli Lagrangian, one 
an easily prove that H is �nite everywhere, C2,
superlinear and stri
tly 
onvex in the �bers. Moreover, under the above assump-
tions, one 
an de�ne a di�eomorphism between TM and T∗M , 
alled the Legendre
transform:
L : TM −→ T∗M
(x, v) 7−→
(x, v)
In parti
ular, L is a 
onjugation between the two �ows (namely the Euler-Lagrange
and Hamiltonian �ows) and
H ◦ L(x, v) =
(x, v), v
− L(x, v) .
Observe that if η is a 1-form on M , then we 
an de�ne a fun
tion on the tangent
spa
e
η̂ : TM −→ R
(x, v) 7−→ 〈η(x), v〉x
and 
onsider a new Tonelli Lagrangian Lη = L − η̂. The asso
iated Hamiltonian
will be Hη(x, p) = H(x, p + η). Moreover, if η is 
losed, then
Ldt and
have the same extremals and therefore the Euler-Lagrange �ows on TM asso
iated
to L and Lη are the same.
Although the extremals are the same, this is not generally true for the minimizers.
What one 
an say is that they stay the same when we 
hange the Lagrangian by
an exa
t 1-form. Thus, for �xed L, the minimizers depend only on the de Rham

ohomology 
lass c = [η] ∈ H1(M ;R). From here, the interest in 
onsidering mod-
i�ed Lagrangians, 
orresponding to di�erent 
ohomology 
lasses.
Let us �x η, a smooth (C2 is enough for what follows) 1-form on M , and let
c = [η] ∈ H1(M ;R) be its 
ohomology 
lass.
As done by Mather in [16℄, it is 
onvenient to introdu
e, for t > 0 and x, y ∈ M ,
the following quantity:
hη,t(x, y) = inf
Lη(γ(s), γ̇(s)) ds ,
where the in�mum is taken over all pie
ewise C1 paths γ : [0, t] −→ M , su
h that
γ(0) = x and γ(t) = y. We de�ne the Peierls barrier as:
hη(x, y) = lim inf
(hη,t(x, y) + α(c)t) ,
where α : H1(M ;R) −→ R is Mather's α fun
tion (see [15℄). It 
an be shown that
this fun
tion is 
onvex and that (only for the autonomous 
ase) the lim inf 
an be
repla
ed by lim.
Observe that hη does not depend only on the 
ohomology 
lass c, but also on the

hoi
e of the representant; namely, if η′ = η+df , then hη′(x, y) = hη(x, y)+ f(y)−
f(x).
Proposition 1. The values of the map hη are �nite. Moreover, the following
properties hold:
i) hη is Lips
hitz;
ii) for ea
h x ∈ M , hη(x, x) ≥ 0;
iii) for ea
h x, y, z ∈ M , hη(x, y) ≤ hη(x, z) + hη(z, y);
iv) for ea
h x, y ∈ M , hη(x, y) + hη(y, x) ≥ 0.
4 ALFONSO SORRENTINO
For a proof of the above 
laims and more, see [16, 10, 8℄. Inspired by these prop-
erties, we 
an de�ne
δc : M ×M −→ R
(x, y) 7−→ hη(x, y) + hη(y, x)
(observe that this fun
tion does a
tually depend only on the 
ohomology 
lass).
This fun
tion is positive, symmetri
 and satis�es the triangle inequality; therefore,
it is a pseudometri
 on
AL,c = {x ∈ M : δc(x, x) = 0} .
AL,c is 
alled the Aubry set (or proje
ted Aubry set) asso
iated to L and c, and
δc is Mather's pseudometri
. In [16℄, Mather has showed that this is a non-empty

ompa
t subset of M , that 
an be Lips
hitzly lifted to a 
ompa
t invariant subset
of TM .
De�nition. The quotient Aubry set (ĀL,c, δ̄c) is the metri
 spa
e obtained by
identifying two points in AL,c, if their δc-pseudodistan
e is zero.
We shall denote an element of this quotient by x̄ = {y ∈ AL,c : δc(x, y) = 0}.
These elements (that are also 
alled c-stati
 
lasses, see [8℄) provide a partition of
AL,c into 
ompa
t subsets, that 
an be lifted to invariant subsets of TM . They are
really interesting from a dynami
al systems point of view, sin
e they 
ontain the α
and ω limit sets of c-minimizing orbits (see [16, 8℄ for more details).
For the sake of our proof, it is 
onvenient to adopt Fathi's weak KAM theory point
of view (we remand the reader to [10℄ for a self-
ontained presentation).
De�nition. A lo
ally lips
hitz fun
tion u : M −→ R is a subsolution ofHη(x, dxu) =
k, with k ∈ R, if Hη(x, dxu) ≤ k for almost every x ∈ M .
This de�nition makes sense, be
ause, by Radema
her's theorem, we know that dxu
exists almost everywhere.
It is possible to show that there exists c[η] ∈ R, su
h that Hη(x, dxu) = k admits
no subsolutions for k < c[η] and has subsolutions for k ≥ c[η]. The 
onstant c[η] is

alled Mañé's 
riti
al value and 
oin
ides with α(c), where c = [η] (see [8℄).
De�nition. u : M −→ R is a η-
riti
al subsolution, if Hη(x, dxu) ≤ α(c) for almost
every x ∈ M .
Denote by Sη the set of 
riti
al subsolutions. This set Sη is non-empty. In fa
t,
Fathi showed (see [10℄) that:
Proposition 2. If u : M −→ R is a η-
riti
al subsolution, then for every x, y ∈ M
u(y)− u(x) ≤ hη(x, y) .
Moreover, for any x ∈ M , the fun
tion hη,x(·) := hη(x, ·) is a η-
riti
al subsolution.
Using this result, he provided a ni
e representation of hη, in terms of the η-
riti
al
subsolutions.
Corollary 1. If x ∈ AL,c and y ∈ M ,
hη(x, y) = sup
(u(y)− u(x)) .
This supremum is a
tually attained.
Proof. It is 
lear, from the proposition above, that
hη(x, y) ≥ sup
(u(y)− u(x)) .
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 5
Let us show the other inequality. In fa
t, sin
e hη,x is a η-
riti
al subsolution and
x ∈ AL,c (i.e., hη(x, x) = 0), then
hη(x, y) = hη,x(y)− hη,x(x) ≤ sup
(u(y)− u(x)) .
This shows that the supremum is attained. ✷
This result 
an be still improved. Fathi and Si
onol� proved in [11℄:
Theorem (Fathi, Si
onol�). For any η-
riti
al subsolution u : M −→ R and for
ea
h ε > 0, there exists a C1 fun
tion ũ : M −→ R su
h that:
i) ũ(x) = u(x) and Hη(x, dxũ) = α(c) on AL,c;
ii) |ũ(x) − u(x)| < ε and Hη(x, dxũ) < α(c) on M \ AL,c.
In parti
ular, this implies that C1 η-
riti
al subsolutions are dense in Sη with the
uniform topology. This result has been re
ently improved by Patri
k Bernard (see
[5℄), showing that every η-
riti
al subsolution 
oin
ides, on the Aubry set, with a
C1,1 η-
riti
al subsolution.
Denote the set of C1 η-
riti
al subsolutions by S1η and the set of C1,1 η-
riti
al
subsolutions by S1,1η .
Corollary 2. For x, y ∈ AL,c, the following representation holds:
hη(x, y) = sup
u∈S1η
(u(y)− u(x)) = sup
u∈S1,1η
(u(y)− u(x)) .
Moreover, these suprema are attained.
It turns out to be 
onvenient, to 
hara
terize the elements of ĀL,c (i.e., the c-
quotient 
lasses) in terms of η-
riti
al subsolutions.
Let us 
onsider the following set:
Dc = {u− v : u, v ∈ Sη}
(it depends only on the 
ohomology 
lass c and not on η) and denote by D1c and
D1,1c , the sets 
orresponding, respe
tively, to C1 and C1,1 η-
riti
al subsolutions.
Proposition 3. For x, y ∈ AL,c,
δc(x, y) = sup
(w(y) − w(x)) = sup
w∈D1c
(w(y) − w(x)) =
= sup
w∈D1,1c
(w(y)− w(x))
and this suprema are attained.
Proof. From the de�nition of δc(x, y), we immediately get:
δc(x, y) = hη(x, y) + hη(y, x) =
= sup
(u(y)− u(x)) + sup
(v(x) − v(y)) =
= sup
u,v∈Sη
[(u(y)− v(y))− (u(x)− v(x))] =
= sup
(w(y) − w(x)) .
The other equalities follow from the density results we mentioned above. ✷
Proposition 4. If w ∈ Dc, then dxw = 0 on AL,c. Therefore AL,c ⊆
w∈D1,1c Crit(w) ,
where Crit(w) is the set of 
riti
al points of w.
Proof. This is an immediate 
onsequen
e of a result by Fathi (see [10℄); namely, if
u, v ∈ Sη, then they are di�erentiable on AL,c and dxu = dxv. ✷
6 ALFONSO SORRENTINO
Proposition 5. If w ∈ Dc, then it is 
onstant on any quotient 
lass of ĀL,c;
namely, if x, y ∈ AL,c and δc(x, y) = 0, then w(x) = w(y).
Proof. From the representation formula above, it follows that:
0 = δc(x, y) = sup
w̃∈Dc
(w̃(y)− w̃(x)) ≥ w(y) − w(x)
0 = δc(y, x) = sup
w̃∈Dc
(w̃(x)− w̃(y)) ≥ w(x) − w(y) .
For any w ∈ D1c , let us de�ne the following evaluation fun
tion:
ϕw : (ĀL,c, δ̄c) −→ (R, | · |)
x̄ 7−→ w(x) .
• ϕw is well de�ned, i.e., it does not depend on the element of the 
lass at
whi
h w is evaluated;
• ϕw(ĀL,c) = w(AL,c) ⊆ w(Crit(w));
• ϕw is Lips
hitz, with Lips
hitz 
onstant 1. In fa
t:
ϕw(x̄)− ϕw(ȳ) = w(x) − w(y) ≤ δc(x, y) = δ̄c(x̄, ȳ)
ϕw(ȳ)− ϕw(x̄) = w(y)− w(x) ≤ δc(y, x) = δ̄c(ȳ, x̄) .
Therefore:
|ϕw(x̄)− ϕw(ȳ)| ≤ δ̄c(x̄, ȳ) .
As we shall see, these fun
tions play a key role in the proof of our result.
3. The main result.
Our main goal is to show that, under suitable hypotheses on L, there is a well
spe
i�ed 
ohomology 
lass cL, for whi
h (ĀL,cL , δ̄cL) is totally dis
onne
ted, i.e.,
every 
onne
ted 
omponent 
onsists of a single point.
Consider L : TM −→ R a Tonelli Lagrangian and the asso
iated Legendre trans-
L : TM −→ T∗M
(x, v) 7−→
(x, v)
Remember that T∗M , as a 
otangent bundle, may be equipped with a 
anoni
al
symple
ti
 stru
ture. Namely, if (U , x1, . . . , xd) is a lo
al 
oordinate 
hart for
M and (T∗U , x1, . . . , xd, p1, . . . , pd) the asso
iated 
otangent 
oordinates, one 
an
de�ne the 2-form
dxi ∧ dpi .
It is easy to show that ω is a symple
ti
 form (i.e., it is non-degenerate and 
losed).
In parti
ular, one 
an 
he
k that ω is intrinsi
ally de�ned, by 
onsidering the 1-form
on T∗U
pi dxi ,
whi
h satis�es ω = −dλ and is 
oordinate-indipendent; in fa
t, in terms of the
natural proje
tion
π : T∗M −→ M
(x, p) 7−→ x
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 7
the form λ may be equivalently de�ned pointwise without 
oordinates by
λ(x,p) = (dπ(x,p))
∗p ∈ T∗(x,p)T∗M .
The 1-form λ is 
alled the Liouville form (or the tautologi
al form).
Consider now the se
tion of T∗M given by
ΛL = L(M × {0}) =
(x, 0)
: x ∈ M

orresponding to the 1-form
ηL(x) =
(x, 0) · dx =
(x, 0) dxi .
We would like this 1-form to be 
losed, that is equivalent to ask ΛL to be a
Lagrangian submanifold, in order to 
onsider its 
ohomology 
lass cL = [ηL] ∈
H1(M ;R). Observe that this 
ohomology 
lass 
an be de�ned in a more intrinsi
way; in fa
t, 
onsider the proje
tion
π|ΛL : ΛL ⊂ T∗M −→ M ;
this indu
es an isomorphism between the 
ohomology groupsH1(M ;R) andH1(ΛL;R).
The preimage of [λ|ΛL ] under this isomorphism is 
alled the Liouville 
lass of ΛL
and one 
an easily show that it 
oin
ides with cL.
We 
an de�ne the set:
L(M) = {L : TM −→ R : L is a Tonelli Lagrangian and ΛL is Lagrangian} .
This set is non-empty and 
onsists of Lagrangians of the form
L(x, v) = f(x) + 〈η(x), v〉x +O(‖v‖2),
with f ∈ C2(M) and η a C2 
losed 1-form on M . In parti
ular, it in
ludes the
me
hani
al Lagrangians, i.e., Lagrangians of the form
L(x, v) =
‖v‖2x + U(x) ,
namely the sum of the kineti
 energy and a potential U : M −→ R. More gen-
erally, it 
ontains the symmetri
al (or reversible) Lagrangians, i.e., Lagrangians
L : TM −→ R su
h that
L(x, v) = L(x,−v) ,
for every (x, v) ∈ TM .
In fa
t, in the above 
ases,
(x, 0) ≡ 0; therefore ΛL = M × {0} (the zero se
tion
of the 
otangent spa
e), that is 
learly Lagrangian, and cL = 0.
We 
an now state our main result:
Main Theorem. Let M be a 
ompa
t 
onne
ted manifold of dimension d ≥ 1 and
let L ∈ L(M) be a Lagrangian su
h that L(x, 0) ∈ Cr(M), with r ≥ 2d − 2 and
(x, 0) ∈ C2(M). Then, the quotient Aubry set (ĀL,cL , δ̄cL), 
orresponding to
the Liouville 
lass of ΛL, is totally dis
onne
ted, i.e., every 
onne
ted 
omponent

onsists of a single point.
This result immediately implies:
Corollary 3 (Symmetri
al Lagrangians). Let M be a 
ompa
t 
onne
ted manifold
of dimension d ≥ 1 and let L(x, v) be a symmetri
al Tonelli Lagrangian on TM ,
su
h that L(x, 0) ∈ Cr(M), with r ≥ 2d−2. Then, the quotient Aubry set (ĀL,0, δ̄0)
is totally dis
onne
ted.
8 ALFONSO SORRENTINO
More spe
i�
ally,
Corollary 4 (Me
hani
al Lagrangians). Let M be a 
ompa
t 
onne
ted manifold
of dimension d ≥ 1 and let L(x, v) = 1
‖v‖2x + U(x) be a me
hani
al Lagrangian
on TM , su
h that the potential U ∈ Cr(M), with r ≥ 2d − 2. Then, the quotient
Aubry set (ĀL,0, δ̄0) is totally dis
onne
ted.
Remark. This result is optimal, in the sense of the regularity of the potential U ,
for ĀL,0 to be totally dis
onne
ted. In fa
t, Mather provided in [19℄ examples of
quotient Aubry sets isometri
 to the unit interval, 
orresponding to me
hani
al
Lagrangians L ∈ C2d−3,1−ε(TTd), for any 0 < ε < 1.
Before proving the main theorem, it will be useful to show some useful results.
Lemma 1. Let us 
onsider L ∈ L(M), su
h that ∂L
(x, 0) ∈ C2(M), and let H be
the asso
iated Hamiltonian.
(1) Every 
onstant fun
tion u ≡ const is a ηL-
riti
al subsolution. In parti
u-
lar, all ηL-
riti
al subsolutions are su
h that dxu ≡ 0 on AL,cL.
(2) For every x ∈ M ,
(x, 0) =
(x, ηL(x)) = 0 .
Proof.
(1) The se
ond part follows immediatly from the fa
t that, if u, v ∈ SηL , then
they are di�erentiable on AL,cL and dxu = dxv (see [10℄).
Let us show that u ≡ const is a ηL-
riti
al subsolution; namely, that
HηL(x, 0) ≤ α(cL)
for every x ∈ M . It is su�
ient to observe:
• HηL(x, 0) = −L(x, 0); in fa
t:
HηL(x, 0) = H(x, ηL(x)) = H
(x, 0)
(x, 0), 0
− L(x, 0) =
= −L(x, 0) .
• let v be dominated by LηL + α(cL) (see [10℄, for the existen
e of su
h
fun
tions), i.e., for ea
h 
ontinuous pie
ewise C1 
urve γ : [a, b] −→ M
we have
v(γ(b))− v(γ(a)) ≤
LηL(γ(t), γ̇(t)) dt + α(cL)(b − a) .
Then, 
onsidering the 
onstant path γ(t) ≡ x, one 
an easily dedu
e
α(cL) ≥ sup
(−LηL(x, 0)) = − inf
LηL(x, 0) ;
therefore,
α(cL) ≥ −LηL(x, 0) = −L(x, 0) = HηL(x, 0)
for every x ∈ M .
(2) The inverse of the Legendre transform 
an be written in 
oordinates
L−1 : T∗M −→ TM
(x, p) 7−→
(x, p)
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 9
Therefore,
(x, 0) = L−1 (L(x, 0)) = L−1
(x, 0)
= L−1((x, ηL(x))) =
(x, ηL(x)
In parti
ular, observing that for any ηL-
riti
al subsolution u, HηL(x, dxu) = α(cL)
on AL,cL , we 
an easily dedu
e from above that:
AL,cL ⊆ {L(x, 0) = −α(cL)} = {H(x, ηL(x)) = α(cL)}
α(cL) = sup
(−L(x, 0)) = − inf
L(x, 0) =: e0 ,
as denoted in [14, 7℄.
Let us observe that in general
e0 ≤ min
c∈H1(M ;R)
α(c) = −β(0) ,
where β : H1(M ;R) −→ R is Mather's β-fun
tion, i.e., the 
onvex 
onjugate of
α (in [14, 7℄, the right-hand-side quantity is referred to as stri
t 
riti
al value).
Therefore, we are 
onsidering an extremal 
ase in whi
h e0 = α(cL) = minα(c); it
follows also quite easily that cL ∈ ∂β(0), namely, it is a subgradient of β at 0.
A 
ru
ial step in the proof of our result will be the following lemma, that 
an be
read as a sort of relaxed version of Sard's Lemma (the proof will be mainly based
on the one in [1℄).
Main Lemma. Let U ∈ Cr(M), with r ≥ 2d − 2, be a non-negative fun
tion,
vanishing somewhere and denote A = {U(x) = 0}. If u : M −→ R is C1 and
satis�es ‖dxu‖2x ≤ U(x) in an open neighborhood of A, then |u(A)| = 0 (where | · |
denotes the Lebesgue measure in R).
See se
tion 4 for its proof.
In parti
ular, it implies this essential property.
Corollary 5. Under the hypotheses of the main theorem, if u ∈ SηL , then
|u(AL,cL)| = 0
(where | · | denotes the Lebesgue measure in R).
Proof (Corollary). First of all, we 
an assume that u ∈ S1ηL , be
ause of Fathi
and Si
onol�'s theorem. By Taylor's formula, it follows that there exists an open
neighborhood W of AL,cL , su
h that for all x ∈ W :
α(cL) ≥ HηL(x, dxu) = HηL(x, 0) +
(x, 0) · dxu+
(1− t)∂
(x, t dxu)(dxu)
2 dt .
Let us observe the following.
• From the previous lemma, one has that
(x, 0) = 0 ,
for every x ∈ M .
10 ALFONSO SORRENTINO
• From the stri
t 
onvexity hypothesis, it follows that there exists γ > 0 su
h
that:
(x, t dxu)(dxu)
2 ≥ γ‖dxu‖2x
for all x ∈ M and 0 ≤ t ≤ 1.
Therefore, for x ∈ W :
α(cL) ≥ HηL(x, dxu) ≥ HηL(x, 0) +
‖dxu‖2x =
= −L(x, 0) + γ
‖dxu‖2x .
The assertion will follow from the previous lemma, 
hoosing
U(x) =
(α(cL) + L(x, 0)) .
In fa
t, U ∈ Cr, with r ≥ 2d − 2, by hypothesis; moreover, it satis�es all other

onditions, be
ause
α(cL) = − inf
L(x, 0)
AL,cL ⊆ {x ∈ W : L(x, 0) = −α(cL)} = {x ∈ W : U(x) = 0} =: A .
For, the previous lemma allows us to 
on
lude that
|u(AL,cL)| = 0 .
Proof (Main Theorem). Suppose by 
ontradi
tion that ĀL,cL is not totally dis-

onne
ted; therefore it must 
ontain a 
onne
ted 
omponent Γ with at least two
points x̄ and ȳ. In parti
ular
δ̄c(x̄, ȳ) = hηL(x, y) + hηL(y, x) > 0,
for some x ∈ x̄ and y ∈ ȳ; therefore, we have hηL(x, y) > 0 or hηL(y, x) > 0. From
the representation formula for hηL , it follows that there exists u ∈ S1,1ηL ⊆ D
(sin
e u = u−0, and v = 0 is a ηL-
riti
al subsolution), su
h that |u(y)−u(x)| > 0.
This implies that the set ϕu(Γ) is a 
onne
ted set in R with at least two di�erent
points, hen
e it is a non degenerate interval and its Lebesgue measure is positive.
ϕu(Γ) ⊆ ϕu(ĀL,cL) = u(AL,cL)
and 
onsequently
∣ϕu(Γ)
∣ ≤ |u(AL,cL)|.
This 
ontradi
ts the previous 
orollary. ✷
In parti
ular, this proof suggests a possible approa
h to generalize the above result
to more general Lagrangians and other 
ohomology 
lasses.
De�nition. A C1 fun
tion f : M −→ R is of Morse-Sard type if |f(Crit(f))| = 0,
where Crit(f) is the set of 
riti
al points of f and | · | denotes the Lebesgue measure
in R.
Proposition 6. Let M be a 
ompa
t 
onne
ted manifold of dimension d ≥ 1, L
a Tonelli Lagrangian and c ∈ H1(M ;R). If ea
h w ∈ D1,1c is of Morse-Sard type,
then the quotient Aubry set (ĀL,c, δ̄c) is totally dis
onne
ted.
This proposition and Sard's lemma (see [4℄) easily imply Mather's result in dimen-
sion d ≤ 2 (autonomous 
ase); it su�
es to noti
e that Sard's lemma (in dimension
d) holds for Cd−1,1 fun
tions.
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 11
Corollary 6. Let M be a 
ompa
t 
onne
ted manifold of dimension d ≤ 2. For
any L Tonelli Lagrangian and c ∈ H1(M ;R), the quotient Aubry set (ĀL,c, δ̄c) is
totally dis
onne
ted.
Remark. The main problem be
omes now to understand under whi
h 
onditions
on L and c, these di�eren
es of subsolutions are of Morse-Sard type. Unfortunately,
one 
annot use the 
lassi
al Sard's lemma, due to a la
k of regularity of 
riti
al
subsolutions: in general they will be at most C1,1. In fa
t, although it is always
possible to smooth them up out of the Aubry set and obtain fun
tions in C∞(M \
AL,c)∩C1,1(M), the presen
e of the Aubry set (where the value of their di�erential
is pres
ribed) represents an obsta
le that it is impossible to over
ome. It is quite
easy to 
onstru
t examples that do not admit C2 
riti
al subsolutions: just 
onsider
a 
ase in whi
h AL,c is all the manifold and it is not a C1 graph. For instan
e, this
is the 
ase if M = T and H(x, p) = 1
)2 − sin2(πx); in fa
t, there is only one

riti
al subsolution (up to 
onstants), that turns out to be a solution (AL, 2
= T),
and it is given by a primitive of sin(πx)− 2
; this is 
learly C1,1 but not C2.
On the other hands, the above results suggest that, in order to prove the Morse-Sard
property, one 
ould try to 
ontrol the 
omplexity of these fun
tions (à la Yomdin),
using the rigid stru
ture provided by Hamilton-Ja
obi equation and the smoothness
of the Hamiltonian, rather than the regularity of the subsolutions. There are several
di�
ulties in pursuing this approa
h in the general 
ase, mostly related to the nature
of the Aubry set. We hope to understand these �spe
ulations� more in depth in the
future.
4. Proof of the Main Lemma.
De�nition. Consider a fun
tion f ∈ Cr(Rd). We say that f is s - �at at x0 ∈ Rd
(with s ≤ r), if all its derivatives, up to the order s, vanish at x0.
The proof of the main lemma is based on the following version of Kneser-Glaeser's
Rough 
omposition theorem (see [1, 20℄).
Proposition 7. Let V, W ⊂ Rd be open sets, A ⊂ V , A∗ ⊂ W 
losed sets.
Consider U ∈ Cr(V ), with r ≥ 2, a non-negative fun
tion that is s-�at on A ⊂
{U(x) = 0}, with s ≤ r − 1, and g : W −→ V a Cr−s fun
tion, with g(A∗) ⊂ A.
Then, for every open pre-
ompa
t set W1 ⊃ A∗ properly 
ontained in W , there
exists
F : Rd −→ R
satisfying the following properties:
i) F ∈ Cr−1(Rd);
ii) F ≥ 0;
iii) F (x) = U(g(x)) = 0 on A∗;
iv) F is s-�at on A∗;
v) {F (x) = 0} ∩W1 = A∗;
vi) there exists a 
onstant K > 0, su
h that U(g(x)) ≤ KF (x) on W1.
See se
tion 5 for its proof.
To prove the main lemma, it will be enough to show that for every x0 ∈ M , there
exists a neighborhood Ω su
h that it holds. For su
h a lo
al result, we 
an assume
that M = U is an open subset of Rd, with x0 ∈ U . In the sequel, we shall identify
T∗U with U × Rd and for x ∈ U , we identify T∗xU = {x} × Rd. We equip U × Rd
with the natural 
oordinates (x1, . . . , xd, p1, . . . , pd).
12 ALFONSO SORRENTINO
Before pro
eeding in the proof, let us point out that it is lo
ally possible to repla
e
the norm obtained by the Riemannian metri
, by a 
onstant norm on R
Lemma 2. For ea
h 0 < α < 1 and x0 ∈ M , there exists an open neighborhood Ω
of x0, with Ω ⊂ U and su
h that
(1− α)‖p‖x0 ≤ ‖p‖x ≤ (1 + α)‖p‖x0 ,
for every p ∈ T∗xU ∼= Rd and ea
h x ∈ Ω.
Proof. By 
ontinuity of the Riemannian metri
, the norm ‖p‖x tends uniformly
to 1 on {p : ‖p‖x0 = 1}, as x tends to x0. Therefore, for x near to x0 and every
p ∈ Rd \ {0}, we have:
(1− α) ≤
‖p‖x0
≤ (1 + α).
We 
an now prove the main result of this se
tion.
Proof ( Main Lemma). By 
hoosing lo
al 
harts and by lemma 2, we 
an assume
that U ∈ Cr(Ω), with Ω open set in Rd, A = {x ∈ Ω : U(x) = 0} and u : Ω −→ R
is su
h that ‖dxu‖2 ≤ βU(x) in Ω, where β is a positive 
onstant.
De�ne, for 1 ≤ s ≤ r:
Bs = {x ∈ A : U is s - �at at x}
and observe that
A = B1 := {x ∈ A : DU(x) = 0} .
We shall prove the lemma by indu
tion on the dimension d. Let us start with the
following 
laim.
Claim. If s ≥ 2d− 2, then |u(Bs)| = 0.
Proof. Let C ⊂ Ω be a 
losed 
ube with edges parallel to the 
oordinate axes. We
shall show that |u(Bs ∩C)| = 0. Sin
e Bs 
an be 
overed by 
ountably many su
h

ubes, this will prove that |u(Bs)| = 0.
Let us start observing that, by Taylor's theorem, for any x ∈ Bs ∩C and y ∈ C we
U(y) = Rs(x; y),
where Rs(x; y) is Taylor's remainder. Therefore, for any y ∈ C
U(y) = o(‖y − x‖s) .
Let λ be the length of the edge of C. Choose an integer N > 0 and subdivide C in
Nd 
ubes Ci with edges
, and order them so that, for 1 ≤ i ≤ N0 ≤ Nd, one has
Ci ∩Bs 6= ∅. Hen
e,
Bs ∩ C =
Bs ∩ Ci.
Observe that for every ε > 0, there exists ν0 = ν0(ε) su
h that, if N ≥ ν0, x ∈
Bs ∩ Ci and y ∈ Ci, for some 0 ≤ i ≤ N0, then
U(y) ≤ ε
4β(dλ2)d
‖y − x‖s .
Fix ε > 0. Choose xi ∈ Bs ∩ Ci and 
all yi = u(xi). De�ne, for N ≥ ν0, the
following intervals in R:
, yi +
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 13
Let us show that, if N is su�
iently big, then u(Bs ∩ C) ⊂
i=1 Ei.
In fa
t, if x ∈ Bs ∩ C, then there exists 1 ≤ i ≤ N0, su
h that x ∈ Bs ∩ Ci.
Therefore,
|u(x)− yi| = |u(x)− u(xi)| =
= ‖dxu(x̃)‖ · ‖x− xi‖ ≤
βU(x̃)‖x− xi‖ ≤
4β(dλ2)d
‖x̃− xi‖
2 ‖x− xi‖ ≤
2(dλ2)
‖x− xi‖
2(dλ2)
where x̃ is a point in the segment joining x and xi. Sin
e by hypothesis s ≥ 2d− 2,
≥ d. Hen
e, assuming that N > max{λ
d, ν0}, one gets
|u(x)− yi| ≤
and 
an dedu
e the in
lusion above.
To prove the 
laim, it is now enough to observe:
|u(Bs ∩ C)| ≤
|Ei| ≤
≤ εN0
≤ εNd 1
= ε .
From the arbitrariness of ε, the assertion follows easily. ✷
This 
laim immediately implies that u(B2d−2) has measure zero.
In parti
ular, this proves the 
ase d = 1 (sin
e in this 
ase 2d− 2 = 0) and allows
us to start the indu
tion.
Suppose to have proven the result for d− 1 and show it for d. Sin
e
A = (B1 \B2) ∪ (B2 \B3) ∪ . . . ∪ (B2d−3 \B2d−2) ∪B2d−2 ,
it remains to show that |u(Bs \Bs+1)| = 0 for 1 ≤ s ≤ 2d− 3 ≤ r − 1.
Claim. Every x̃ ∈ Bs \Bs+1 has a neighborhood Ṽ , su
h that
|u((Bs \Bs+1) ∩ Ṽ )| = 0 .
Sin
e Bs\Bs+1 
an be 
overed by 
ountably many su
h neighborhoods, this implies
that u(Bs \Bs+1) has measure zero.
Proof. Choose x̃ ∈ Bs \ Bs+1. By de�nition of these sets, all partial derivatives
of order s of U vanish at this point, but there is one of order s + 1 that does not.
Assume (without any loss of generality) that there exists a fun
tion
w(x) = ∂i1∂i2 . . . ∂isU(x)
su
h that
w(x̃) = 0 but ∂1w(x̃) 6= 0 .
14 ALFONSO SORRENTINO
De�ne
h : Ω −→ Rd
x 7−→ (w(x), x2, . . . , xd) ,
where x = (x1, x2, . . . , xd). Clearly, h ∈ Cr−s(Ω) and Dh(x̃) is non-singular;
hen
e, there is an open neighborhood V of x̃ su
h that
h : V −→ W
is a Cr−s isomorphism (with W = h(V )).
Let V1 be an open pre
ompa
t set, 
ontaining x̃ and properly 
ontained in V , and
de�ne A = Bs ∩ V1, A∗ = h(A) and g = h−1. If we 
onsider W1, any open set

ontaining A∗ and properly 
ontained in W , we 
an apply proposition 7 and dedu
e
the existen
e of F : Rd −→ R satisfying properties i)-vi).
De�ne
Ŵ = {(x2, . . . , xd) ∈ Rd−1 : (0, x2, . . . , xd) ∈ W1}
Û(x2, . . . , xd) = C F (0, x2, . . . , xd),
where C is a positive 
onstant to be 
hosen su�
iently big. Observe that Û ∈
Cr−1(Rd−1).
Moreover, property v) of F and the fa
t that A∗ = h(A) ⊆ {0} × Ŵ imply that:
∗ = {0} × B̂1 ,
where B̂1 = {(x2, . . . xd) ∈ Ŵ : F (0, x2, . . . , xd) = 0}. Denote
Â := {(x2, . . . , xd) ∈ Ŵ : Û = 0} = B̂1
and de�ne the following fun
tion on Ŵ :
û(x2, . . . , xd) = u(g(0, x2, . . . , xd)).
We want to show that these fun
tions satisfy the hypotheses for the (d − 1)-
dimensional 
ase. In fa
t:
• Û ∈ Cr−1(Rd−1), with r − 1 ≥ 2d− 3 > 2(d− 1)− 2;
• û ∈ C1(Ŵ ) (sin
e g is in Cr−s(W ), where 1 ≤ s ≤ r − 1);
• if we denote by µ = supW1 ‖dxg‖ < +∞ (sin
e g is C
on W1), then we
have that for every point in Ŵ :
‖dû(x2, . . . , xd)‖2 ≤ ‖dxu(g(0, x2, . . . , xd))‖2‖dxg(0, x2, . . . , xd)‖2 ≤
≤ µ2‖dxu(g(0, x2, . . . , xd))‖2 ≤
≤ βµ2U(g(0, x2, . . . , xd)) ≤
≤ βµ2KF (0, x2, . . . , xd) ≤
≤ Û(x2, . . . , xd),
if we 
hoose C > βµ2K, where K is the positive 
onstant appearing in
proposition 7, property vi).
Therefore, it follows from the indu
tive hypothesis, that:
|û(Â)| = 0.
Sin
e,
u(Bs ∩ V1) ⊆ u(A) = u(g(A∗)) = u(g({0} × B̂1)) =
= û(B̂1) = û(Â) ,
de�ning Ṽ = V1, we may 
on
lude that
|u(Bs ∩ Ṽ )| ≤ |û(Â)| = 0 .
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 15
This 
ompletes the proof of the Main Lemma. ✷
5. Proof of a modified version of Kneser-Glaeser's Rough

omposition theorem.
Now, let us prove proposition 7. We shall mainly follow the presentation in [1℄,
adapted to our needs.
Proof (Proposition 7). Let us start, de�ning a family of polynomials. Supposing
that g is Cr and using the s-�atness hypothesis, we have, for x ∈ A∗ and k =
0, 1, . . . , r :
fk(x) = D
k(U ◦ g)(x) =
s<q≤k
qU(g(x))Di1g(x) . . . Diqg(x) ,(1)
where the se
ond sum is over all the q-tuples of integers i1, . . . , iq ≥ 1 su
h that
i1 + . . .+ iq = k, and σk = σk(i1, . . . , iq).
The 
ru
ial observation is that (1) makes sense on A∗, even when g is Cr−s smooth
(in fa
t ij ≤ k − q + 1 ≤ r − s).
We would like to pro
eed in the fashion of Whitney's extension theorem, in order
to �nd a smooth fun
tion F su
h that DkF = fk on A
, and satisfying the stated

onditions.
Remark. Note that, without any loss of generality, we 
an assume that W is

ontained in an open ball of diameter 1. The general 
ase will then follow from this
spe
ial one, by a straightforward partition of unity argument.
Let us start with some te
hni
al lemmata.
Lemma 3. For x, x′, x0 ∈ A∗ and k = 0, . . . , r, we have:
i≤r−k
fk+i(x)
(x′ − x)i +Rk(x, x′) ,
|Rk(x, x′)|
‖x′ − x‖r−k −→ 0
as x, x′ −→ x0 in A∗.
The proof of this lemma appears without any major modi�
ation in [1℄ (on pages
36-37).
De�ne, for x ∈ A∗ and y ∈ Rd
P (x, y) =
i=s+1
fi(x)
(y − x)i
and its k-th derivative
Pk(x, y) =
i≤r−k
fi+k(x)
(y − x)i .
Lemma 4. For x ∈ A∗ and y ∈ W1,
U(g(y)) = P (x, y) +R(x, y) ,
where |R(x, y)| ≤ C‖y − x‖r.
16 ALFONSO SORRENTINO
Proof. The proof follows the same idea of lemma 3. By Taylor's formula for U ,
U(g(y)) =
q=s+1
DqU(g(x))
(g(y)− g(x))q + I(g(x), g(y))(g(x) − g(y))r .
Obviously,
|I(g(x), g(y))(g(x) − g(y))r| ≤ C1‖y − x‖r ,
therefore it is su�
ient to estimate the �rst term.
Observe that:
g(y) = g(x) +
Dig(x)(y − x)i + J(x, y)(y − x)r−s .
Hen
e, the �rst term in the sum above be
omes:
q=s+1
DqU(g(x))
Dig(x)(y − x)i + J(x, y)(y − x)r−s
k=s+1
ak(y − x)k + R̂(x, y) =
= P (x, y) + R̂(x, y) ,
sin
e
s+1≤q≤k
DqU(g(x))Di1g(x) . . . Diqg(x) =
fk(x)
The remainder terms 
onsist of:
• terms 
ontaining (y − x)k, with k > r;
• terms of the binomial produ
t, 
ontaining J(x, y)(y − x)r−s. They are of
the form:
. . . (y − x)(r−s)j+
where αi ≥ 0 and
αi = q − j. Sin
e q ≥ s+ 1 and s ≤ r − 1, then:
(r − s)j +
iαi ≥ (r − s)j +
= (r − s)j + q − j =
= rj − sj + q − j ≥
≥ rj − (s+ 1)j + s+ 1 =
= r + r(j − 1)− (s+ 1)(j − 1) =
= r + (r − s− 1)(j − 1) ≥ r .
Therefore, for x ∈ A∗ and y ∈ W1
R̂(x, y)
≤ C2‖y − x‖r ,
and the lemma follows taking C = C1 + C2. ✷
Next step will 
onsist of 
reating a Whitney's partition. We will start by 
overing
W1 \ A∗ with an in�nite 
olle
tion of 
ubes Kj, su
h that the size of ea
h Kj is
roughly proportional to its distan
e from A∗.
First, let us �x some notation. We shall write a ≺ b instead of �there exists a
positive real 
onstant M , su
h that a ≤ Mb � and a ≈ b as short for a ≺ b and
b ≺ a.
Let λ = 1
; this 
hoi
e will 
ome in handy later. For any 
losed 
ube K (with
edges parallel to the 
oordinate axes), Kλ will denote the (1 + λ) - dilation of K
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 17
about its 
enter.
Let ‖ · ‖ be the eu
lidean metri
 on Rd and
d(y) = d(y,A∗) = inf{‖y − x‖ : x ∈ A∗} .
If {Kj}j is the sequen
e of 
losed 
ubes de�ned below, with edges of length ej , let
dj be its distan
e from A
, i.e.,
dj = d(A
,Kj) = inf{‖y − x‖ : x ∈ A∗, y ∈ Kj} .
One 
an show the following 
lassi
al lemma (see for instan
e [1℄ for a proof).
Lemma 5. There exists a sequen
e of 
losed 
ubes {Kj}j with edges parallel to the

oordinate axes, that satis�es the following properties:
i) the interiors of the Kj 's are disjoint;
ii) W1 \A∗ ⊂
iii) ej ≈ dj ;
iv) ej ≈ d(y) for all y ∈ Kλj ;
v) ej ≈ d(z) for all z ∈ W1 \ A∗, su
h that the ball with 
enter z and radius
d(z) interse
ts Kλj ;
vi) ea
h point of W1 \A∗ has a neighborhood interse
ting at most N of the Kλj ,
where N is an integer depending only on d.
Now, let us 
onstru
t a partition of unity on W1 \ A∗. Let Q be the unit 
ube

entered at the origin. Let η be a C∞ bump fun
tion de�ned on Rd su
h that
η(y) =
1 for y ∈ Q
0 for y 6∈ Qλ
and 0 ≤ η ≤ 1. De�ne
ηj(y) = η
y − cj
where cj is the 
enter of Kj and ej is the length of its edge, and 
onsider
σ(y) =
ηj(y) .
Then, 1 ≤ σ(y) ≤ N for all y ∈ W1 \ A∗. Clearly, for ea
h k = 0, 1, 2, . . . we have
that Dkηj(y) ≺ e−kj , for all y ∈ W1 \A∗. Hen
e, by properties iv) and vi) of lemma
5, we have that for ea
h k = 0, 1, . . . , r:
ηj(y) ≺ d(y)−k for all y ∈ W1 \A∗
Dkσ(y) ≺ d(y)−k for all y ∈ W1 \A∗ .
De�ne
ϕj(y) =
ηj(y)
These fun
tions satisfy the following properties:
i) ea
h ϕj is C
and supported on Kλj ;
ii) 0 ≤ ϕj(y) ≤ 1 and
j ϕj(y) = 1, for all y ∈ W1 \A∗;
iii) every point of W1 \ A∗ has a neighborhood on whi
h all but at most N of
the ϕj 's vanish identi
ally;
iv) for ea
h k = 0, 1, . . . , r, Dkϕj(y) ≺ d(y)−k for all y ∈ W1 \ A∗; namely,
there are 
onstants Mk su
h that D
kϕj(y) ≤ Mkd(y)−k;
v) there is a 
onstant α and points xj ∈ A∗, su
h that:
‖xj − y‖ ≤ αd(y), whenever ϕj(y) 6= 0 .
This follows from properties iii) and iv) of lemma 5.
18 ALFONSO SORRENTINO
We 
an now 
onstru
t our fun
tion F . Observe that, from lemma 4:
0 ≤ U(g(y)) = P (xj , y) +R(xj , y) ≤ P (xj , y) + C‖y − xj‖r ;
therefore P (xj , y) ≥ −C‖y − xj‖r.
First, de�ne
P̂j(y) = P (xj , y) + 2C‖y − xj‖r
where C is the same 
onstant as in lemma 4; for what said above,
P̂j(y) ≥ C‖y − xj‖r > 0 in W1 \ {xj} .(2)
Hen
e, 
onstru
t F in the following way:
F (y) =
0 y ∈ A∗
ϕj(y)P̂j(y) y ∈ Rd \A∗ .
We 
laim that this satis�es all the stated properties i)-vi). In parti
ular, properties
ii), iii) and v) follow immediately from the de�nition of F and (2). Moreover,
F ∈ C∞(Rd \ A∗). We need to show that DkF = fk (for k = 0, 1, . . . , r − 1) on
∂A∗ (namely, the boundary of A∗) and that Dr−1F is 
ontinuous on it. The main
di�
ult in the proof, is that DkF is expressed as a sum 
ontaining terms
Dk−mϕj(y)Pm(xj , y),
where ϕj(y) 6= 0. Even if y is 
lose to some x0 ∈ A∗, it 
ould be 
loser to A∗ and
hen
e the bound given by property iv) of ϕj might be
ome large. One 
an over
ome
this problem by 
hoosing a point x∗ ∈ A∗, so that ‖x∗ − y‖ is roughly the same as
d(y) and hen
e, xj is 
lose to x
Lemma 6. For every η > 0, there exists δ > 0 su
h that for all y ∈ W1 \ A∗,
x, x∗ ∈ A∗ and x0 ∈ ∂A∗, we have
‖Pk(x, y)− Pk(x∗, y)‖ ≤ η d(y)r−k ≤ η‖y − x0‖r−k,
whenever k ≤ r and
‖y − x‖ < αd(y)
‖y − x∗‖ < αd(y)
‖y − x0‖ < δ ,
where α is the same 
onstant as in v) above.
See [1℄ (on page 126) for its proof.
Lemma 7. For every η > 0, there exist 0 < δ < 1 and a 
onstant E, su
h that for
all y ∈ W1 \A∗, x∗ ∈ A∗ and x0 ∈ ∂A∗, we have
‖DkF (y)− Pk(x∗, y)‖ ≤ E d(y)r−k ≤ η d(y)r−k−1,
whenever k ≤ r − 1 and
‖y − x∗‖ < αd(y)
‖y − x0‖ < δ .
Proof. Let
Sj,k(x
∗, y) = ∂kP̂j(y)− Pk(x∗, y) .
From lemma 6 (with η = ε, to be de�ned later) and the de�nition of P̂j , we get:
‖Sj,k(x∗, y)‖ ≤ ‖∂kP̂j(y)− Pk(xj , y)‖+ ‖Pk(xj , y)− Pk(x∗, y)‖ ≤
≤ Ckd(y)r−k + εd(y)r−k =
= (Ck + ε)d(y)
r−k .
Then,
F (y)− P (x∗, y) =
ϕj(y)Sj,0(x
∗, y)
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 19
and hen
e
DkF (y)− Pk(x∗, y) =
Dk−iϕj(y)Sj,i(x
∗, y) .
Therefore, 
hoosing ε su�
iently small:
‖DkF (y)− Pk(x∗, y)‖ ≤
‖Dk−iϕj(y)‖ · ‖Sj,i(x∗, y)‖ ≤
Mk−id(y)
−k+i(Ck + ε)d(y)
r−i ≤
≤ E d(y)r−k ≤ η d(y)r−k−1 .
Lemma 8. For every η > 0, there exist 0 < δ < 1 su
h that, for all y ∈ W1 \ A∗,
x∗ ∈ A∗ and x0 ∈ ∂A∗, we have
‖Pk(x∗, y)− Pk(x0, y)‖ ≤ η‖y − x0‖r−k ,
whenever k ≤ r and
‖y − x∗‖ < αd(y)
‖y − x0‖ < δ .
Proof. The proof goes as the one of lemma 6, observing that ‖x∗ − x0‖ ≤ (1 +
α)‖y − x0‖ and
Pk(x0, y)− Pk(x∗, y) =
q≤r−k
Rk+q(x
∗, x0)
(y − x)q .
Claim. For every x0 ∈ ∂A∗ and k = 0, 1, . . . , r − 1:
F (x0) = fk(x0) .
Moreover, Dr−1F is 
ontinuous at x0 ∈ ∂A∗.
This 
laim follows easily from the lemmata above (see [1℄, on page 128, for more
details).
This proves that F ∈ Cr−1(Rd) and 
ompletes the proof of i) and iv).
It remains to show that property vi) holds, namely that there exists a 
onstant
K > 0, su
h that U(g(x)) ≤ KF (x) on W1. Obviously, this holds at every point in
A∗, for every 
hoi
e of K (sin
e both fun
tions vanish there).
Claim. There exists a 
onstant K > 0, su
h that U◦g
≤ K on W1 \A∗.
Proof. Sin
e F > 0 on W1 \ A∗, it is su�
ient to show that U◦gF is uniformly
bounded by a 
onstant, as d(y) goes to zero.
Let us start observing that, for y ∈ Kλj ,
P̂j(y) ≥ C‖y − xj‖r ≥ Cd(y)r ;
therefore:
F (y) =
ϕj(y)P̂j(y) ≥
ϕj(y)Cd(y)
= Cd(y)r .
20 ALFONSO SORRENTINO
Moreover, if x∗ ∈ A∗ su
h that d(y) = ‖y − x∗‖, lemma 4 and 7 imply:
|U(g(y))− F (y)| ≤ |U(g(y))− P (x∗, y)|+ |P (x∗, y)− F (y)| ≤
≤ Cd(y)r + Ed(y)r = (C + E)d(y)r .
Hen
e,
U(g(y))
F (y)
U(g(y))− F (y) + F (y)
F (y)
≤ 1 + |U(g(y))− F (y)|
F (y)
≤ 1 + (C + E)d(y)
Cd(y)r
≤ 2 + E
=: K̃ .
This proves property vi) and 
on
ludes the proof of the proposition. ✷
A
knowledgements. I wish to thank John Mather for having introdu
ed me to this
area and suggested this problem. I am very grateful to him and to Albert Fathi for
their interest and for several helpful dis
ussions.
Referen
es
[1℄ Ralph Abraham and Joel Robbin. Transversal mappings and �ows. An appendix by Al Kelley.
W. A. Benjamin, In
., New York-Amsterdam, 1967.
[2℄ Patri
e Assouad. Plongements lips
hitziens dans R
. Bull. So
. Math. Fran
e, 111(4):429�
448, 1983.
[3℄ Vi
tor Bangert. Mather sets for twist maps and geodesi
s on tori. In Dynami
s reported, Vol.
1, volume 1 of Dynam. Report. Ser. Dynam. Systems Appl., pages 1�56. Wiley, Chi
hester,
1988.
[4℄ Sean M. Bates. Toward a pre
ise smoothness hypothesis in Sard's theorem. Pro
. Amer.
Math. So
., 117(1):279�283, 1993.
[5℄ Patri
k Bernard. Existen
e of C

riti
al sub-solutions of the Hamilton-Ja
obi equation on

ompa
t manifolds. (To appear on Ann. S
i. É
ole Norm. Sup. (4)).
[6℄ D. Burago, S. Ivanov, and B. Kleiner. On the stru
ture of the stable norm of periodi
 metri
s.
Math. Res. Lett., 4(6):791�808, 1997.
[7℄ Gonzalo Contreras, Jorge Delgado, and Renato Iturriaga. Lagrangian �ows: the dynami
s of
globally minimizing orbits. II. Bol. So
. Brasil. Mat. (N.S.), 28(2):155�196, 1997.
[8℄ Gonzalo Contreras and Renato Iturriaga. Global minimizers of autonomous Lagrangians. 22
Colóquio Brasileiro de Matemáti
a. [22nd Brazilian Mathemati
s Colloquium℄. Instituto de
Matemáti
a Pura e Apli
ada (IMPA), Rio de Janeiro, 1999.
[9℄ Albert Fathi. Sard, Whitney, Assouad and Mather. Talk at Re
ent and Future developments
in Hamiltonian Systems, Institut Henri Poin
aré, Paris (Fran
e), May 2005.
[10℄ Albert Fathi. Weak KAM theorem and Lagrangian dynami
s. Cambridge Univ. Press (To
appear).
[11℄ Albert Fathi and Antonio Si
onol�. Existen
e of C

riti
al subsolutions of the Hamilton-
Ja
obi equation. Invent. Math., 155(2):363�388, 2004.
[12℄ Giovanni Forni and John N. Mather. A
tion minimizing orbits in Hamiltonian systems. In
Transition to 
haos in 
lassi
al and quantum me
hani
s (Monte
atini Terme, 1991), volume
1589 of Le
ture Notes in Math., pages 92�186. Springer, Berlin, 1994.
[13℄ Piotr Hajlasz. Whitney's example by way of Assouad's embedding. Pro
. Amer. Math. So
.,
131(11):3463�3467 (ele
troni
), 2003.
[14℄ Ri
ardo Mañé. Lagrangian �ows: the dynami
s of globally minimizing orbits. Bol. So
. Brasil.
Mat. (N.S.), 28(2):141�153, 1997.
[15℄ John N. Mather. A
tion minimizing invariant measures for positive de�nite Lagrangian sys-
tems. Math. Z., 207(2):169�207, 1991.
ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 21
[16℄ John N. Mather. Variational 
onstru
tion of 
onne
ting orbits. Ann. Inst. Fourier (Grenoble),
43(5):1349�1386, 1993.
[17℄ John N. Mather. A property of 
ompa
t, 
onne
ted, laminated subsets of manifolds. Ergodi
Theory Dynam. Systems, 22(5):1507�1520, 2002.
[18℄ John N. Mather. Total dis
onne
tedness of the quotient Aubry set in low dimensions. Comm.
Pure Appl. Math., 56(8):1178�1183, 2003. Dedi
ated to the memory of Jürgen K. Moser.
[19℄ John N. Mather. Examples of Aubry sets. Ergodi
 Theory Dynam. Systems, 24(5):1667�1723,
2004.
[20℄ Hassler Whitney. Analyti
 extensions of di�erentiable fun
tions de�ned in 
losed sets. Trans.
Amer. Math. So
., 36(1):63�89, 1934.
Department of Mathemati
s, Prin
eton University, Prin
eton (NJ), 08544-1000 U.S.
E-mail address: asorrent�math.prin
eton.edu
	1. Introduction.
	2. The Aubry set and the quotient Aubry set.
	3. The main result.
	4. Proof of the Main Lemma.
	5. Proof of a modified version of Kneser-Glaeser's Rough composition theorem.
	References
