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Singular measures for class P-circle homeomorphisms with several break points

Published online by Cambridge University Press:  30 November 2012

ABDELHAMID ADOUANI  and
HABIB MARZOUGUI
ABDELHAMID ADOUANI
Affiliation:
University of Carthage, Faculty of Science of Bizerte, Department of Mathematics, Zarzouna, 7021, Tunisia (email: arbi.abdelhamid@gmail.com; habib.marzouki@fsb.rnu.tn, hmarzoug@ictp.it)
HABIB MARZOUGUI
Affiliation:
University of Carthage, Faculty of Science of Bizerte, Department of Mathematics, Zarzouna, 7021, Tunisia (email: arbi.abdelhamid@gmail.com; habib.marzouki@fsb.rnu.tn, hmarzoug@ictp.it)
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Abstract

Let f be a class P-homeomorphism of the circle with break point singularities, that is, differentiable except at some singular points where the derivative has a jump. Let f have irrational rotation number and Df be absolutely continuous on every continuity interval of Df. We prove that if the product of the f-jumps along any subset of break points is distinct from 1 then the invariant measure μf is singular with respect to the Haar measure. This result generalizes previous results obtained by Dzhalilov and Khanin, Dzhalilov, Akhadkulov, Dzhalilov–Liousse and Mayer. Moreover, we prove that if the rotation number ρ(f) is irrational of bounded type then (a) if the product of the f-jumps on some orbit is distinct from 1 then the invariant measure μf is singular with respect to the Haar measure m, and (b) if the product of the f-jumps on each orbit is equal to 1 and D2f∈Lp (S1) for some p>1 then μfis equivalent to the Haar measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 423 - 456
Copyright
©2012 Cambridge University Press 

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References

[1]Adouani, A. and Marzougui, H.. Sur les homéomorphismes du cercle de classe P C r par morceaux ( r>1) qui sont conjugués C r par morceaux aux rotations irrationnelles. Ann. Inst. Fourier 58 (2008), 755–775.Google Scholar
[2]Adouani, A. and Marzougui, H.. On piecewise smoothness of conjugacy of class P circle homeomorphisms to diffeomorphisms and rotations. Dyn. Syst. 27 (2012), 169–186.CrossRefGoogle Scholar
[3]Akhadkulov, Kh. A.. Some circle homeomorphisms with break-type singularities. Russian Math. Surveys 61(5) (2006), 981–983.Google Scholar
[4]de Melo, W. and van Strien, S.. One-Dimensional Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete, 25). Springer, Berlin, 1993.Google Scholar
[5]Denjoy, A.. Sur les courbes définies par les équations differentielles à la surface du tore. J. Math. Pures Appl. 11 (1932), 333–375.Google Scholar
[6]Dzhalilov, A. and Khanin, K. M.. On invariant measure for homeomorphisms of a circle with a break point. Funct. Anal. Appl. 32 (1998), 153–161.Google Scholar
[7]Dzhalilov, A.. Piecewise smoothness of conjugate homeomorphisms of a circle with corners. Theoret. Math. Phys. 120 (1999), 961–972.Google Scholar
[8]Dzhalilov, A., Liousse, I. and Mayer, D.. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete Contin. Dyn. Syst. 24 (2009), 381–403.Google Scholar
[9]Finzi, A.. Sur le problème de la génération d’une transformation donnée d’une courbe fermée par une transformation infinitésimale. Ann. Sci. Ec. Norm. Super., 3e série 67 (1950), 243–305.Google Scholar
[10]Hasselblatt, B. and Katok, A.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.Google Scholar
[11]Hewitt, E. and Stromberg, K.. Real and Abstract Analysis (Graduate text in Mathematics 25). Springer, Berlin, 1965.Google Scholar
[12]Herman, M.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5–234.CrossRefGoogle Scholar
[13]Katznelson, Y. and Ornstein, D.. The differentiability of the conjugation of certain diffeomorphisms of the circle. Ergod. Th. & Dynam. Sys. 9 (1989), 643–680.Google Scholar
[14]Katznelson, Y.. Sigma-finite invariant measures for smooth mappings of the circle. J. Anal. Math. 31 (1977), 1–18.Google Scholar
[15]Khanin, K. M. and Sinai, Y. G.. Smoothness of conjugacies of diffeomorphisms of the circle with rotations. Russian Math. Surveys 44 (1989), 69–99.Google Scholar
[16]Liousse, I.. Nombre de rotation, mesures invariantes et ratio set des homéomorphismes affines par morceaux du cercle. Ann. Inst. Fourier 55 (2005), 1001–1052.Google Scholar
[17]Navas, A.. Groups of Circle Diffeomorphisms (Chicago Lectures in Mathematics). University of Chicago Press, Chicago, IL, 2011.Google Scholar
[18]Poincaré, H.. Oeuvres complètes t.1 (1885), 137–158.Google Scholar
[19]Sinai, Y. G.. Topics in Ergodic Theory. Princeton University Press, Princeton, NJ, 1994.Google Scholar
[20]Stark, J.. Smooth conjugacy and renormalisation for diffeomorphisms of the circle. Nonlinearity 1 (1988), 541–575.Google Scholar
[21]Swiatek, G.. Rational rotation numbers for maps of the circle. Comm. Math. Phys. 119 (1988), 109–128.Google Scholar
[22]Teplinskii, A.. On cross-ratio distortion and Schwartz derivative. Nonlinearity 21 (2008), 2777–2783.Google Scholar
[23]Yoccoz, J. C.. Il n’y a pas de contre-exemple de Denjoy analytique. C. R. Acad. Sci. 298(serie I) (1984), 141–144.Google Scholar