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On commuting circle mappings and simultaneous Diophantine approximations. (English) Zbl 0689.58031
In this paper the question is studied when a family of orientation- preserving, pairwise commuting circle diffeomorphisms are simultaneously smoothly conjugate to rotations. For the local problem, i.e. for \(C^{\infty}\)-diffeomorphisms sufficiently close to rotations a sufficient condition is given in terms of their rotation numbers. This condition is related to the nature of the simultaneous approximation of the rotation numbers by rationals with the same denominator. These questions lead to a problem in the theory of simultaneous Diophantine approximations which is solved in Theorem 2. The corresponding global problem, which is the analogue of Herman’s celebrated theorem on a single circle mapping, remains an open problem.
Reviewer: Jürgen Moser

MSC:
37E10 Dynamical systems involving maps of the circle
37E45 Rotation numbers and vectors
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
11J13 Simultaneous homogeneous approximation, linear forms
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