On piecewise class \(PC^r\) homeomorphisms of the circle which are piecewise \(C^ r (r\geq 1)\) conjugate to irrational rotations
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 \textit{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 \textit{A. A. Dzhalilov} and \textit{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 \(\mathcal{P}^r(S^1).\)
scientific article; zbMATH DE number 5298320