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These functions satisfy (34) which completes the proof. D. 1: We start with an integrable truncation of f . Let fˆ be defined by (1) with Pj , Qj ≡ 0. Then, the time 1 map of the Hamiltonian 1 Br, r 2 coincides with f0 , as an easy calculation shows. Construct a time periodic deformation {H(·, t)}t∈T of H0 so that the time 1 map of H(·, t) coincides with f and so that H(·, t) is C s+1 close to H0 (·) near r = 0 for all t ∈ T. Since H0 is convex in r, s ≥ 2, and we are interested in small r, it implies the desired convexity of H(·, t) in r for each t ∈ T.

If ρ(µ) = h, we also say that µ is h-minimal. Denote by Mh closure of the union of supports of all h-minimal measures from PL . This set Mh ⊂ T Tn × T is called Mather set. By the above lemma Mh is always nonempty. A probability measure µ ∈ PL is c-minimal for c ∈ H 1 (Tn , R), if it minimizes Ac (µ) = A(µ) − ρ(µ), c over all invariant probability measures. Ac (µ) as above is called c-action of a measure. 2. — For every cohomology class c ∈ H 1 (Tn , R) there exists a c-minimal probability measure µ ∈ PL such that A(µ) < ∞.

2. — For every cohomology class c ∈ H 1 (Tn , R) there exists a c-minimal probability measure µ ∈ PL such that A(µ) < ∞. Denote by Mc closure of supports of the union of all c-minimal measures from PL . Mc ⊂ T Tn × T is also called Mather set. By the above lemma Mc is always nonempty. Mather [Ma] proved that ∪h∈H1 (Tn ,R) Mh = ∪c∈H 1 (Tn ,R) Mc . It turns out that Mc can be “nicely” projected onto the base Tn × T. Graph Theorem. Let π : TTn × T → Tn × T be the natural projection onto the base. e.

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