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

Abstract

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 piecewise linear (PL)-homeomorphism or a piecewise quadratic homeomorphism.

For a PL-homeomorphism f having the property (ii): f is conjugate to a rotation either by a PL-homeomorphism or by a piecewise analytic homeomorphism.

Keywords:

• piecewise C ¹ class P-homeomorphism
• PL-homeomorphism
• rotation number
• conjugacy
• break point
• jump

AMS Subject Classification:

• Primary: 37C15
• 37E10

Acknowledgements

The second author thanks the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, where part of this work was done, for their hospitality.

Notes

ICTP Regular Associate.

Additional information

Notes on contributors

Habib Marzougui

ICTP Regular Associate.