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Differentiability of the minimal average action as a function of the rotation number. (English) Zbl 0766.58033
The author considers the class of all \(f\)-invariant measures with angular rotation number \(w\), provided the generic \(f\) being a finite composition of exact twist diffeomorphisms. The main purpose of the paper is to study the dependence of the minimal average acton \(A(w)\) over the above class of measures upon the variable \(w\). The author proves the differentiability of \(A(w)\) at every irrational number \(w\). He shows that in general the function \(A(w)\) is not differentiable at any rational point \(w\). The criterion of differentiability of \(A(w)\) at a fixed point \(w=p/q\) is formulated. The paper contains a brief review of results on the subject and also a discussion of some open problems for which the author’s techniques provide the solution.

MSC:
37A99 Ergodic theory
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37C55 Periodic and quasi-periodic flows and diffeomorphisms
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[1] Aubry, S.,The Devil’s Staircase Transformation in Incommensurate Lattices inThe Riemann Problem, Complete Integrability, and Arithmetic Applications, ed. by Chudnovsky and Chudnovsky, Lecture Notes in Math925 (1982), 221-245. Springer-Verlag.
[2] Bangert, V.,Mather Sets for Twist Maps and Geodesics on Tori, Dynamics Reported1 (1988), 1-54. · Zbl 0664.53021
[3] Mather, J.N.,Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology21 (1982), 457-467. · Zbl 0506.58032
[4] ?,Modulus of continuity for Peierls’s barrier inPeriodic Solutions of Hamiltonian Systems and Related Topics edited by P.H. Rabinowitz, et al., NATO ASI Series C: vol. 209. Dordrecht: D. Reidel (1987), 177-202
[5] ?,Destruction of Invariant Circles, Ergodic Theory and Dynamical Systems8 (1988), 199-214. · Zbl 0688.58024
[6] ?,Minimal Measures, Comm. Math. Helv.64 (1989), 375-394. · Zbl 0689.58025
[7] Mather, J.N.,Action Minimizing Invariant Measures for Positive Definite Lagrangian Systems. preprint, ETH, 1989, 60pp · Zbl 0850.70195
[8] Mather, J.N.,Variational Construction of Orbits of Twist Diffeomorphisms. preprint, ETH, 1990, 73pp · Zbl 0737.58029
[9] ?,More Denjoy minimal sets area preserving diffeomorphisms, Comm. Math. Helv.60 (1985), 508-557. · Zbl 0597.58015
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