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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 \(\mathcal{P}^r(S^1).\)

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
Full Text: DOI Numdam EuDML
[1] Adouani, A.; Marzougui, H., Conjugacy of piecewise \(C^1\)-homeomorphisms of class \(P\) of the circle · Zbl 1254.37017
[2] Denjoy, A., Sur LES courbes définies par LES équations différentielles à la surface du tore, J. Math. Pures Appl., 11, 333-375, (1932) · Zbl 0006.30501
[3] Dzhalilov, A.; Khanin, K. M., On invariant measure for homeomorphisms of a circle with a break point, Functional Analysis and its Applications, 32, 3, 153-161, (1998) · Zbl 0921.58035
[4] Dzhalilov, A. A., Piecewise smoothness of conjugate homeomorphisms of a circle with corners, Theoretical and Mathematical Physics, 120, 2, 961-972, (1999) · Zbl 0988.37046
[5] Dzhalilov, A. A.; Liousse, I., Circle homeomorphisms with two break points, Nonlinearity, 19, 1951-1968, (2006) · Zbl 1147.37025
[6] Fayad, B.; Khanin, K., Smooth linearization of commuting circle diffeomorphisms · Zbl 1177.37045
[7] Katznelson, Y., Sigma-finite invariant measures for smooth mappings of the circle, J. Analyse Math., 31, 1-18, (1977) · Zbl 0346.28012
[8] Katznelson, Y.; Ornstein, D., The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dynam. Systems, 9, 4, 681-690, (1989) · Zbl 0819.58033
[9] Katznelson, Y.; Ornstein, D., The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergod.Th. and Dynam.Sys., 9, 4, 643-680, (1989) · Zbl 0819.58033
[10] Khanin, K. M.; Sinaĭ, Ya. G., Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Russian Math. Surveys, 44, 1, 69-99, (1989) · Zbl 0701.58053
[11] Liousse, I., Nombres de rotation dans LES groupes de Thompson généralisé, automorphismes
[12] Liousse, I., Nombre de rotation, mesures invariantes et ratio set des homéomorphismes affines par morceaux du cercle, Ann. Inst. Fourier, Grenoble, 55, 2, 1001-1052, (2005) · Zbl 1079.37033
[13] Liousse, I., PL homeomorphisms of the circle which are piecewise \(C^1\) conjugate to irrational rotations, Bull Braz Math Soc, New Series, 35, 2, 269-280, (2005) · Zbl 1136.37333
[14] Minakawa, H., Classification of exotic circles of \({\rm PL}_+ (S^1),\) Hokkaido Math. J., 26, 3, 685-697, (1997) · Zbl 0896.57024
[15] Moser, J., On commuting circle mapping and simultaneous Diophantine approximations, Math. Z., 205, 105-121, (1990) · Zbl 0689.58031
[16] Poincaré, H., Oeuvres complètes, (1885)
[17] Stark, J., Smooth conjugacy and renormalisation for diffeomorphisms of the circle, Nonlinearity, 1, 4, 541-575, (1988) · Zbl 0725.58040
[18] Yoccoz, J.-C., Il n’ y a pas de contre-exemple de Denjoy analytique, C.R.A.S., 298, 7, 141-144, (1984) · Zbl 0573.58023
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