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PL homeomorphisms of the circle which are piecewise \(C^1\) conjugate to irrational rotations. (English) Zbl 1136.37333

Summary: For a PL homeomorphism \(f\) with irrational rotation number \(\alpha\), the following properties are equivalent:
(i) \(f\) is conjugate to the rotation by \(\alpha\) through a piecewise \(C^1\) homeomorphism,
(ii) the number of break points of \(f^n\) is bounded by some constant that doesn’t depend on \(n\),
(iii) \(f\) is conjugate to an affine 2-intervals exchange transformation (with rotation number \(\alpha\)) through a PL homeomorphism,
(iv) \(f\) is conjugate to the rotation by \(\alpha\) through a piecewise analytic homeomorphism.

MSC:

37E10 Dynamical systems involving maps of the circle
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
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