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Preprints Archive: Abstract of IC2010029 (2010)

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On piecewise smoothness of conjugacy of class P circle homeomorphisms to diffeomorphisms and rotations

by Abdelhamid Adouani and Habib Marzougui

Document info: Pages 20, Figures 1.

We give a characterization of piecewise $C^{1}$ class $P$ homeomorphism $f$ of the circle with irrational rotation number and finitely many break points which is piecewise $C^{1}$ conjugate to a $C^{1}$-diffeomorphism. The following properties are equivalent: (i) $f$ is conjugate to a $C^{1}$-diffeomorphism of the circle by a piecewise $C^{1}$ homeomorphism. (ii) the product of jumps of $f$ in the break points contained in a same orbit is trivial. (iii) $f$ is conjugate to a $C^{1}$-diffeomorphism of the circle by a $PL$ homeomorphism or a piecewise quadratic homeomorphism. For a $PL$-homeomorphism $f$ having the property (ii): $f$ is conjugate to a rotation by either a $PL$ omeomorphism or a piecwise analytic homeomorphism.

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