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

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**Additional info for Instability of resonant totally elliptic points of symplectic maps in dimension 4**

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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.