A new proof of the AubryMather's theorem.
01 Dec 1992Mathematische Zeitschrift (Springer Science and Business Media LLC)Vol. 210, Iss: 1, pp 441448
About: This article is published in Mathematische Zeitschrift.The article was published on 19921201 and is currently open access. It has received 25 citation(s) till now. The article focuses on the topic(s): Hilbert cube & Rotation number.
Topics: Hilbert cube (65%), Rotation number (53%)
Summary (1 min read)
Jump to: [A new proof of the AubryMather’s theorem] and [3. Lyapunov functions for non rest points.]
A new proof of the AubryMather’s theorem
 The authors present a new proof of the theorem of Aubry and Mather on the existence of quasi periodic orbits for monotone twist maps of the cylinder.
 Whereas Aubry had to take sequences of periodic orbits of a “good” (“Birkhoff”) type to obtain his quasiperiodic ones, Mather worked in a certain functional space that picked up the orbits of a chosen, rational or irrational, rotation number.
 This paper was written while the author was on a postdoctoral at the ETH in Zürich.
 The following proof , due to Angenent, was already given in [G2], Lemma 1.22, and the authors include it for the convenience of the reader.
 The authors let the reader show that if the operator solution of the linearised equation: (2.2) u̇ = −HessW (x(t))u(t) is strictly positive, then the flow is strictly monotone.
3. Lyapunov functions for non rest points.
 All the terms in the computation above being invariant under x→ τ0,1x, the lemma is also proven for x in Yω/Z.
 The next lemma will show how to use the “Lyapunov” functions WN to find rest points for the flow.
 Let C be compact invariant for the flow ζt, also known as proof.
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A New Proof of the AubryMather's Theorem A New Proof of the AubryMather's Theorem
Christophe Golé
Smith College
, cgole@smith.edu
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A new proof of the AubryMather’s theorem
Christophe Gol´e †
Abstract:We present a new proof of the theorem of Aubry and Mather on the existence of quasi
periodic orbits for monotone twist maps of the cylinder. The method uses Aubry’s discrete
setting, but works directly with sequences of irrational rotation number, avoiding to take limits
of periodic orbits.
Key words: Twist maps, quasiperiodic orbits, AubryMather sets.
0. Introduction
The maps that we consider here are monotone (positive) twist maps of the cylinder
A = (R/Z) × R. These maps are C
1
diﬀeomorphisms that are area preserving and have
“zero ﬂux”: the algebraic area enclosed between a loop and its image by the map is zero.
They also satisfy a twist condition, which enables one to set up a discrete variational
problem.
To understand such maps, according to Poincar´e who ﬁrst studied them, one should
ﬁrst understand periodic orbits. They can be of diﬀerent homotopy type, depending on
how much they turn around the circle component of A, and the length of their period. The
quotient of these two numbers gives the rotation number of the orbit which is rational. A
variation on the theorem of Poincar´e and Birkhoﬀ then asserts that there exists (at least
two) periodic orbits of all (prime) rational rotation number. In [G1], we proved an analog
to this theorem for symplectic twist maps (called monotone maps there) of T
n
× R
n
.
It was not until the late seventies that the existence of orbits of all rotation numbers
was proved. This was done independently by Aubry [AL] and Mather [M], with methods
quite diﬀerent from one another. Whereas Aubry had to take sequences of periodic orbits
of a “good” (“Birkhoﬀ”) type to obtain his quasiperiodic ones, Mather worked in a certain
functional space that picked up the orbits of a chosen, rational or irrational, rotation
number.
The proof that we present here uses the sequence space in which Aubry’s variational
calculus was set, but we are able to restrict ourselves apriori to a subspace of sequences
having a prescribed rotation number.
Apart from what we think is a simpliﬁcation of the existing proofs ( to the exclusion
of that of Angenent in [An 2]), we hope that this method may generalise to symplectic
twist maps, where the main problem until now was to deﬁne what Birkhoﬀ periodic orbits
should mean in this context. Of course, we do use this notion in this paper, but some new,
key steps (section 3) are valid for higher dimensional maps. We have tried to make this
paper as self contained as we could, and included theorems on the energy ﬂow that had
either been stated or even proved before (section 2).
Note again that we are working here in the unbounded annulus, or cylinder. Restrict
ing oneself to a bounded annulus is quite possible with our method, but to the price of
unrewarding complications in the notation. It would also be easy to extend this method to
ﬁnite compositions of twist maps of the same sign, by adding their generating functions.
† SUNY at Stony Brook
1
The author would like to thank Prof. E. Zehnder for very useful conversations and
his encouragement to write this paper, as well as Prof. J. Moser and the referee for their
helpful comments. This paper was written while the author was on a postdoctoral at the
ETH in Z¨urich. The author would like to thank all the staﬀ for their help.
1.Twist maps and their energy ﬂow.
Let F be a diﬀeomorphism of A = (R/Z) × R. Denote by Ω = dy ∧ dx the canonical
symplectic form on A (seen as the cotangent bundle of the circle), x being the angular
variable, y the ﬁber variable. Then Ω = dα, where α = ydx.
We write F (x, y) = (X, Y ). To say that F is area preserving means:
(1.1) F
∗
Ω − Ω = dY ∧ dX − dy ∧ dx = 0
(i.e., F is symplectic). To say that it preserves the ﬂux means:
(1.2) F
∗
α − α = Y dX − ydx = dS
for some real function S on A (i.e., F is exact symplectic). Of course (1.2) implies (1.1).
Finally, the twist condition is given by
(1.3)
∂X
∂y
> 0
If we work in the covering space R
2
of A, keeping the same notation, the twist condition
implies that ψ : (x, y) → (x, X) is a diﬀeomorphism from R
2
onto its image. Here, we will
suppose that ψ is a diﬀeomorphism onto R
2
. In [G1](section 4) we gave some conditions
under which this is true. The standard family, for one , satisﬁes these conditions. The
method we expose can be reproduced in the general case, with appropriate restrictions on
the set of rotation numbers, and with little gain of insight. Because F (x, y) = (X, Y ) ⇒
F (x + 1, y) = (X + 1, Y ) in the covering space, we have that ψ(x + 1, y) = (x + 1, X + 1).
Since ψ is a coordinate change, S can be seen as a function from R
2
to R satisfying the
periodicity condition:
(1.4) S(x + 1, X + 1) = S(x, X).
From (1.2), one can see that S is a generating function for F in the classical mechanic
sense of the term:
y = −∂
1
S(x, X)
Y = ∂
2
S(x, X)
Also, the twist condition (1.3) translates into:
∂
1
∂
2
S(x, X) > 0.
Let z
k
= F
k
(z
0
) = (x
k
, y
k
). The orbit {z
k
} is completely determined by the sequence
{x
k
} of (R)
Z
. Indeed, from (1.2), we deduce:
y
k
= −∂
1
S(x
k
, x
k+1
) = ∂
2
S(x
k−1
, x
k
)
2
This can be written:
∂
1
S(x
k
, x
k+1
) + ∂
2
S(x
k−1
, x
k
) = 0.
This equation can be formally interpreted as:
(1.5)
∇W (x) = 0, for
W (x) =
+∞
X
−∞
S(x
k
, x
k+1
) and x ∈ (R)
Z
.
One can think of the above construction as a discrete version of the classical mechanics
one: the map ψ is the analog to the Legendre transformation (X − x is the discretised
velocity) and equation (1.5) is a formulation of the “least action principle”.
Of course, W is not well deﬁned, since the sum is in general not convergent. However,
“∇W ” is well deﬁned and generates a ﬂow on a subspace of (R)
Z
that we call the energy
ﬂow.
More precisely, we endow R
Z
with the norm :
kxk =
+∞
X
−∞
x
k

2
k
We let X be the subspace of R
Z
of elements of bounded norm, which is a Banach space.
Giving ourselves an ω in R, we deﬁne:
Y
ω
= {x ∈ R
Z
 x
ω
= sup
k∈Z
x
k
− kω < ∞},
on which one can either put the topology induced by the inclusion of Y
ω
in X or the l
∞
topology given by the metric:
x − y
∞
= sup
k∈Z
x
k
− y
k

It is important to notice that elements of Y
ω
have rotation number ω, that is:
x ∈ Y
ω
⇒ lim
k→∞
x
k
k
= ω,
and this deﬁnition coincides with the rotation number of an orbit of F when {x
k
} deﬁnes
the x coordinates of such an orbit.
On R
Z
, we have a Z
2
action given by:
(τ
m,n
x)
k
= x
k+m
+ n
We deﬁne Y
ω
/Z := Y
ω
/τ
0,1
. This will ultimatly be the space on which we will be working.
2. Existence and monotonicity of the ﬂow
3
In this section, we prove the existence of a C
1
energy ﬂow on Y
ω
which is monotone
with respect to the partial order on sequences, and whose rest points correspond to orbits
of F .
Proposition 1: Suppose that the generating function S is C
2
. The inﬁnite system of
O.D.E’s:
(1.6) −∇W (x)
k
= ˙x
k
= −[∂
1
S(x
k
, x
k+1
) + ∂
2
S(x
k−1
, x
k
)]
deﬁnes a C
1
local ﬂow ζ
t
on Y
ω
, for both topologies on Y
ω
. The rest points of ζ
t
on Y
ω
correspond to orbits of the map F with rotation number ω. Furthermore, when S has a
bounded second derivative, the system deﬁnes a C
1
ﬂow on X.
proof: Y
ω
is a Banach manifold diﬀeomorphic to l
∞
: x
k
→ x
k
+ kω gives the diﬀeo
morphism from l
∞
∼
=
Y
0
to Y
ω
, with obvious inverse. We want to show that the map
x → ∇W (x) is a (locally) Lipschitz vector ﬁeld on Y
ω
, and that it has one order of
diﬀerentiability less than that of S. For this we prove the following lemma:
Lemma 1: Let x ∈ Y
ω
. Then the set {(x
k
, x
k+1
)}
k∈Z
/Z is bounded. Hence S and all its
(existing ) derivatives are bounded on this set.
proof: We want to show that (x
k
−E(kω), x
k+1
−E(kω)) is bounded in R
2
, uniformaly
in k, where E is the integer part function. But:
x
k
− E(kω) ≤ x
k
− kω + kω − E(kω) ≤ x
ω
+ 1.
Since E(kω) − E((k + 1)ω) ≤ ω, the same method shows:
x
k+1
− E(kω) ≤ x
ω
+ 1 + ω
We conclude the proof of Lemma 1 reminding the reader that S is periodic (see (1.4)).
Coming back to the proof of the proposition, we notice that lemma 1 shows, among
other things that, when x is in Y
ω
, ∇W (x) is in l
∞
, which is exactly the tangent space to
Y
ω
since the latter is an aﬃne manifold modeled on l
∞
.
To show that ∇W is Lipschitz in the norm k k, we calculate:
k∇W (x) − ∇W (y)k
=
∞
X
−∞
∇W (x)
k
− ∇W (y)
k

2
k
≤ sup
k∈Z,t∈[0,1]
∇
2
S((1 − t)x
k
+ ty
k
, (1 − t)x
k+1
+ ty
k+1
)
∞
X
−∞
(x
k
− y
k
, x
k+1
− y
k+1
)
2
k
≤ 2Kkx − yk,
where K exists since by lemma 1 the second derivative of S is estimated over a bounded
set in the case where both x and y are in Y
ω
. In the case where x and y are only assumed
4
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References
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Abstract: We present a rigorous study of the classical groundstates under boundary conditions of a class of onedimensional models generalizing the discrete FrenkelKontorova model. The extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasiperiodic and chaotic orbits. For all boundary conditions, we select among all the extremum solutions of the energy of the model, those which correspond to the groundstates of the infinite system. We prove that these groundstates are either periodic (commensurate) or quasiperiodic (incommensurate) but are never chaotic. We also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions. The topological structure of the whole set of groundstates is described in details. In addition to physical applications, consequences for twist map homeomorphisms are mentioned. Part II (S. Aubry, P.Y. LeDaeron and G. Andre) will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map.
577 citations
••
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