Home Nonfiction 1 • Instability of resonant totally elliptic points of by Kaloshin V., Mather J.N., Valdinoci E.

Instability of resonant totally elliptic points of by Kaloshin V., Mather J.N., Valdinoci E.

By Kaloshin V., Mather J.N., Valdinoci E.

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The function βL : H1 (Tn , R) → R, βL (h) = A(µ), where µ is an h−minimal probability measure, is called Mather’s β-function. The function αL : H 1 (Tn , R) → R, αL (c) = suph∈H1 (Tn ,R) { h, c − βL (h)} is called Mather’s α-function. 4. — [Ma] Both α-function and β-function are convex and conjugate by the Legendre transform. By definition, βL (h) + αL (c) ≥ h, c , h ∈ H1 (Tn , R), c ∈ H 1 (Tn , R). To distinguish from the standard Legendre transform (6) the map (29) Lβ : H1 (Tn , R) → {compact, convex, non-empty subsets of H 1 (Tn , R)}, defined by letting Lβ (h) be the set of c ∈ H 1 (Tn , R) for which the inequality in (29) becomes equality, is called Fenchel-Legendre transform.

It turns out that this construction is independent of η, provided η is Ma˜ n´e critical and [η]Tn = c. One can show that ρc ≥ 0, satisfies the triangle inequality, and is independent of the choice of a Ma˜ n´e critical closed one form η. The set Ac = {(θ, τ ) : ρc ((θ, τ ), (θ, τ )) = 0} is called Aubry set. One can show [Ma2] that Mc ⊂ Ac ⊂ N c ⊂ T Tn × T for all c ∈ H 1 (Tn , R) and Ac also satisfies the Graph Theorem stated above. Appendix B. 5. — For positive ρ and 0 < α < 1 consider the space of C s,d smooth function on Kρα (θ, r) with the natural C s Whitney topology in Kρα , where m ≤ d ≤ 34 VADIM KALOSHIN, JOHN N.

Denote by X ˜ ˜ t (θ, r) = dft (f˜t )−1 (θ, r) , by i ˜ α the interior derivative of a 1-form α given by X dt Xt ˜ t , and by st (θ, r) a form of generating function, given by (39). Then by X ˜ f (·, t) = i ˜ rdθ − (f˜t−1 )∗ H Xt d st . dt ˜ f (·, t) = −i ˜ dr ∧ dθ. One can check that dH Xt ˜ f (·, t) from time t = 0 to t = 1 equals By the construction, the time map of H f . However, {H(·, t)}t∈[0,1] is not necessarily periodic in t. To attain periodicity we slightly modify the above construction.

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