Variational Methods

Hint. Apply the twisted conjugacy theorem to each H (R + ⋅, ⋅), with R ∈ ℝn close to 0 such that the torus r = R is Diophantine for Ko and, using Proposition 4.4, extend the so-obtained function R ↦ β(R) as a smooth function.

Bibliographical comments.

–It is possible to give a proof, patterned on [87, p. 626], that ψ is C1 without the assumption that ϕ′(x) has unique inverse (or right inverse). Yet the proof simplifies and the estimates improve under the combined two additional assumptions. In particular, the existence of a right inverse of ϕ′(x) makes the inverse ψ unique and, thus, allows us to ignore the way it was built (a posteriori regularity result).

–Latzutkin understood, in the case of the standard map, the fundamental importance of ...