Smith ScholarWorks Smith ScholarWorks
Mathematics and Statistics: Faculty
Publications
Mathematics and Statistics
12-1-1992
A New Proof of the Aubry-Mather's Theorem A New Proof of the Aubry-Mather's Theorem
Christophe Golé
Smith College
, cgole@smith.edu
Follow this and additional works at: https://scholarworks.smith.edu/mth_facpubs
Part of the Mathematics Commons
Recommended Citation Recommended Citation
Golé, Christophe, "A New Proof of the Aubry-Mather's Theorem" (1992). Mathematics and Statistics:
Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/85
This Article has been accepted for inclusion in Mathematics and Statistics: Faculty Publications by an authorized
administrator of Smith ScholarWorks. For more information, please contact scholarworks@smith.edu
A new proof of the Aubry-Mather’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, Aubry-Mather 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 [A-L] 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