Differentiability of the minimal average action as a function of the rotation number.

*(English)*Zbl 0766.58033The 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.

Reviewer: T.F.Filippova (Sverdlovsk)

##### 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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\textit{J. N. Mather}, Bol. Soc. Bras. Mat., Nova Sér. 21, No. 1, 59--70 (1990; Zbl 0766.58033)

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