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On piecewise class \(PC^r\) homeomorphisms of the circle which are piecewise \(C^ r (r\geq 1)\) conjugate to irrational rotations. (Sur les homéomorphismes du cercle de classe \(PC^r\) par morceaux \((r\geq 1)\) qui sont conjugués \(C^ r\) par morceaux aux rotations irrationnelles.) (French) Zbl 1147.37024
It is a well known fact that any orientation-preserving diffeomorphism of the circle \(S^1\) of class \(C^r\), \(r\geq 2,\) having an irrational rotation number \(\rho\) is topologically conjugate to the rotation of angle \(\rho\). The same applies for a more general class of homeomorphisms, the class \(\mathcal{P}\) composed by orientation-preserving homeomorphisms \(f\) which are differentiable except for a countable set of “break points” holding some appropriate conditions of boundedness respect their right and left derivatives. Inside \(\mathcal{P}\) the authors consider the so-called class \(\mathcal{P}^r(S^1), r\geq 1,\) of homeomorphisms \(f\) such that \(f\) is \(C^{[r]}\) (\([\cdot]\) is meant the integer part of \(r\)) except for a finite number of points (singular points) for which it is assumed the existence of right and left derivatives up to the order \(n,\) and such that \(D^{[r]}f\) satisfies a Hölder condition of order \(r-[r]\) in any interval \(I\subset S^1\) where \(f\) is \(C^{[r]}.\)
In the main result of the present paper the authors characterize the elements of \(\mathcal{P}^r(S^1)\) having an irrational rotation number which are topologically conjugate to a \(C^r\)-diffeomorphism via a topological condition, named \(D_r\) (too much involved to be described here), related with the differentiability of certain iterates of \(f\) in the singular points. Moreover, the conjugacy homeomorphism is proved to be inside \(\mathcal{P}^r(S^1)\) and to be a piecewise polynomial.
Additionally, if the rotation number \(\rho\) of \(f\) is Diophantine of order \(\tau \geq 0\) , that is \(| \rho -\frac{p}{q}| >\frac{C}{q^{2+\tau}}\) for any rational number \(\frac{p}{q}\) and some positive constant \(C,\) then \(f\) is topologically conjugate to the rotation \(R_{\rho}\) by a homeomorphism which is piecewise of class \(C^{r-1-\tau\varepsilon}\) for all \(\varepsilon >0\). This result can be viewed as a generalization of another stated in A. A. Dzhalilov [Theor. Math. Phys. 120, No. 2, 961–972 (1999; Zbl 0988.37046)]. On the other hand, by using the above mentioned result of Diophantine nature and considering a result of A. A. Dzhalilov and I. Liousse [Nonlinearity 19, No. 8, 1951–1968 (2006; Zbl 1147.37025)], the authors present sufficient conditions depending on the break points and the property \(D_r\) in order to obtain whether the unique measure \(\mu_f\) preserved by \(f\in \mathcal{P}^r(S^1)\) is singular respect to the Haar’s measure \(m\) or equivalent to \(m.\)
Finally the authors prove that \(\mathcal{P}^r(S^1)\) is a group; and if \(G\) is an abelian subgroup of it and \(G\) possesses at least two elements whose irrational rotation numbers are rationally independent, then \(G\) is conjugate to a subgroup of \(\mathrm{Diff}^{r}_{+}(S^1)\) and the conjugacies are piecewise polynomials of \(\mathcal{P}^r(S^1).\)

MSC:
37E10 Dynamical systems involving maps of the circle
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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