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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
##### Keywords:
twist diffeomorphism; invariant measure; rotation number
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##### References:
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