Conjugacy of Piecewise $C^{1}$-Homeomorphisms of class $P$ of the circle

Abdelhamid Adouani (1) , Habib Marzougui (1)

1 Faculté des Sciences de Bizerte [Université de Carthage]

We give a characterization of piecewise $C^{1}$-homeomorphism of class $P$ of the circle with irrational rotation number and finitely many break points which are piecewise differentiably conjugate to $C^{1}$-diffeomorphims. The following properties are equivalent: i) $f$ is conjugate to a $C^{1}$-diffeomorphism of the circle by a piecewise $C^{1}$-homeomorphism of class $P$. ii) The number of break points of $f^{n}$ is bounded by some constant that doesn't depend on $n$. iii) The product of jumps of $f$ in the break points contained in a same orbit is equal $1$. iv) $f$ is conjugate to a $C^{1}$-diffeomorphism of the circle by a piecewise quadratic homeomorphism of class $P$. This characterization extend Liousse's Theorem for $PL$ homeomorphisms of the circle (\cite{iL04}).

Mots clés: piecewise $C^{1}$-homeomorphism of class $P$; $PL$ homeomorphism; rotation number; conjugacy; break point; jump.; jump

Domaines: Systèmes dynamiques [math.DS]

Abdelhamid Adouani, Habib Marzougui. Conjugacy of Piecewise $C^{1}$-Homeomorphisms of class $P$ of the circle. 2005. ⟨hal-00007995v3⟩