A new proof of the Aubry-Mather's theorem.
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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Citations
31 citations
Additional excerpts
...In this section we prove the existence of solution to the variational equation of a twist potential. The idea is inspired by [ 4 ] and [6]....
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31 citations
Cites methods from "A new proof of the Aubry-Mather's t..."
...Step 2: Existence and Asymptotics of the Parabolic Flow The next lemma for the asymptotics is similar to Lemmas 2 and 3 of Golé (1992)....
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...Step 2: Existence and Asymptotics of the Parabolic Flow The next lemma for the asymptotics is similar to Lemmas 2 and 3 of Golé (1992)....
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...The proof uses a combination of the methods of Mather (1985) and Golé (1992); very roughly, we build a one parameter family of approximate solutions ũD following Mather (1985); then, following Golé (1992), we apply to them the flow of the equation ut = u− Fu x u and show that the closure of uD t ·…...
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...…of Mather (1985) and Golé (1992); very roughly, we build a one parameter family of approximate solutions ũD following Mather (1985); then, following Golé (1992), we apply to them the flow of the equation ut = u− Fu x u and show that the closure of uD t · t≥0 contains a solution of (1); lastly we…...
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29 citations
Cites background or methods from "A new proof of the Aubry-Mather's t..."
...It was shown by Golé [7] that this flow is well-defined on a suitable subspace X ⊂ RZ d of configurations that contains all Birkhoff configurations....
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...In dimension d = 1, the concept of a ghost circle was introduced by Golé....
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...It was provided by Golé in [7] in dimension d = 1....
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...Golé starts his construction by assuming that ω = q p ∈ Q is rational and that an appropriate periodic action function Wp,q(x) is a Morse function....
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...Ghost circles were already constructed for twist maps by Golé [8]....
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References
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