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Journal ArticleDOI

A new proof of the Aubry-Mather's theorem.

01 Dec 1992-Mathematische Zeitschrift (Springer Science and Business Media LLC)-Vol. 210, Iss: 1, pp 441-448

AboutThis article is published in Mathematische Zeitschrift.The article was published on 1992-12-01 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)

A new proof of the Aubry-Mather’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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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
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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
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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 = 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 = , where α = ydx.
We write F (x, y) = (X, Y ). To say that F is area preserving means:
(1.1) F
= 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
k1
, x
k
)
2

This can be written:
1
S(x
k
, x
k+1
) +
2
S(x
k1
, 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
kZ
|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
kZ
|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
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
k1
, 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
)}
kZ
/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
kZ,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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Cites result from "A new proof of the Aubry-Mather's t..."

  • ...The idea of using the gradient semi-flow and its comparison properties for Aubry–Mather theory seems to have originated in [2] and was used to prove the classical result of Aubry–Mather in [40]....

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Book
01 May 2003
Abstract: 1 One-dimensional variational problems.- 1.1 Regularity of the minimals.- 1.2 Examples.- 1.3 The accessory variational problem.- 1.4 Extremal fields for n=1.- 1.5 The Hamiltonian formulation.- 1.6 Exercises to Chapter 1.- 2 Extremal fields and global minimals.- 2.1 Global extremal fields.- 2.2 An existence theorem.- 2.3 Properties of global minimals.- 2.4 A priori estimates and a compactness property.- 2.5 Ma for irrational a, Mather sets.- 2.6 Ma for rational a.- 2.7 Exercises to chapter II.- 3 Discrete Systems, Applications.- 3.1 Monotone twist maps.- 3.2 A discrete variational problem.- 3.3 Three examples.- 3.4 A second variational problem.- 3.5 Minimal geodesics on T2.- 3.6 Hedlund's metric on T3.- 3.7 Exercises to chapter III.- A Remarks on the literature.- Additional Bibliography.

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  • ...In [26] the proofs are made explicit for this situation....

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References
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Journal ArticleDOI
Abstract: We present a rigorous study of the classical ground-states under boundary conditions of a class of one-dimensional models generalizing the discrete Frenkel-Kontorova model. The extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasi-periodic 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 ground-states of the infinite system. We prove that these ground-states are either periodic (commensurate) or quasi-periodic (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 ground-states 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


Journal ArticleDOI
01 Jan 1982-Topology
Abstract: この論文で用いる方法は, Percivalが数値的に準周期軌道を見つけるのに使った方法 [3,4]に 密接に関係している. Percivalは自分の方法で存在定理を証明しなかった. 定理を述べるには,円環自身よりもその普遍被覆Aで仕事する方が都合がよい. A = {(x, y) ∈ R : 0 ≤ y ≤ 1}とする. T : A → Aを平行移動 T (x, y) = (x + 1, y)とする. f はAの面積保 存, 方向保存, かつ境界成分保存の同相写像であって, fT = Tfを満たすとする. 加えて, y > z なら f(x, y)1 > f(x, z)1と仮定する. ここで p = (x, y) ∈ Aのとき p1 = x と約束する. これが 「ねじれ条件」である. fi = fR × i, i = 0, 1と置く. B = {(x, x′) ∈ R : f0(x) ≤ x′ ≤ f1(x)}とする. ねじれ条件 より, 各 (x, x′) ∈ Bに対して唯一の y = g(x, x′) ∈ [0, 1]および y′ = g′(x, x′) ∈ [0, 1]があって, f(x, y) = (x′, y′)を満たす. 関数 gおよび g′はB上の連続関数である. 任意の同相写像 h : R → Rで h(x+ 1) = h(x) + 1を満たすものに対して

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Abstract: : A generalization of the class of monotone twistmaps to maps of S sub 1 x R sub n is proposed. The existence of Birkhoff orbits is studied, and a criterion for positive topological entropy is given. These results are then specialized to the case of monotone twist maps. Finally it is shown that there is a large class of symplectic maps to which the foregoing discussion applies. Keywords: Dynamical systems; Theorems.

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Journal ArticleDOI
S. B. Angenent1
Abstract: We study the oscillation properties of periodic orbits of an area preserving twist map. The results are inspired by the similarity between the gradient flow of the associated action-function, and a scalar parabolic PDE in one space dimension. The Conley-Zehnder Morse theory is used to construct orbits with prescribed oscillatory behavior.

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Abstract: Completely ordered invariant circles are found for the gradient of the energy flow in the state space, containing the critical sets corresponding to the Birkhoff orbits of all rotation number In particular, these ghost circles contain the Aubry-Mather sets and map-invariant circles as completely critical sets when these exist We give a criterion for a sequence of rational ghost circles to converge to a completely critical one

33 citations