Abstract
In this work, a critical circle homeomorphism with several break points is considered. The circle homeomorphism \(f\) with the irrational rotation number \(\rho \) is well known to be strictly ergodic; i.e., it has the unique \(f\)-invariant probability measure \(\mu \). The invariant measure of critical circle homeomorphisms with a finite number of break points is proved to be singular with respect to the Lebesgue measure.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory (Nauka, Moscow, 1980).
A. Denjoy, “Sur les courbes définies par les équations différentielles à la surface du tore,” J. Math. Pures Appl. 11, 333–375 (1932).
M. Herman, “Sur la conjugaison différentiable des difféomorphismes du cercle a des rotations,” Publ. Math. Inst. Hautes Études Sci. 49, 5–233 (1979). https://doi.org/10.1007/bf02684798
A. A. Dzhalilov and K. Khanin, “On an invariant measure for homeomorphisms of a circle with a point of break,” Funct. Anal. Its Appl. 32, 153–161 (1998). https://doi.org/10.1007/BF02463336
A. A. Dzhalilov and I. Liousse, “Circle homeomorphisms with two break points,” Nonlinearity 19, 1951–1968 (2006). https://doi.org/10.1088/0951-7715/19/8/010
A. A. Dzhalilov, I. Liousse, and D. Mayer, “Singular measures of piecewise smooth circle homeomorphisms with two break points,” Discrete Contin. Dyn. Syst. 24, 381–403 (2009). https://doi.org/10.3934/dcds.2009.24.381
A. A. Dzhalilov, D. Mayer, and U. A. Safarov, “Piecewise-smooth circle homeomorphisms with several break points,” Izv. Math. 76, 94–112 (2012). https://doi.org/10.1070/im2012v076n01abeh002576
A. A. Teplinsky, “A circle diffeomorphism with breaks that is absolutely continuously linearizable,” Ergodic Theory Dyn. Syst. 38, 371–383 (2018). https://doi.org/10.1017/etds.2016.24
E. de Faria and P. Guarino, “There are no σ-finite absolutely continuous invariant measures for multicritical circle maps,” Nonlinearity 34, 6727–6749 (2021). https://doi.org/10.1088/1361-6544/ac1a02
J. C. Yoccoz, “Il n’ya pas de contre-exemple de Denjoy analytique,” C. R. Acad. Sci. Paris 298 (7), 141–144 (1984).
J. Graczyk and G. Światek, “Singular measures in circle dynamics,” Commun. Math. Phys. 157, 213–230 (1993). https://doi.org/10.1007/bf02099758
Cinai Ya G, Modern Problems of Ergodic Theory (Fizmatlit, Moscow, 1995).
G. Świątek, “Rational rotation numbers for maps of the circle,” Commun. Math. Phys. 119, 109–128 (1988). https://doi.org/10.1007/BF01218263
C. Preston, Iterates of Maps on an Interval, Lecture Notes in Mathematics, Vol. 999 (Springer, Berlin, 1983). https://doi.org/10.1007/bfb0061749
G. Estevez and E. de Faria, “Real bounds and quasisymmetric rigidity of multicritical circle maps,” Trans. Am. Math. Soc. 370, 5583–5616 (2018). https://doi.org/10.1090/tran/7177
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares that he has no conflicts of interest.
Additional information
Translated by M. Talacheva
About this article
Cite this article
Safarov, U.A. Invariant Measure of a Circle Map with a Mixed Type of Singularities. Russ Math. 67, 59–71 (2023). https://doi.org/10.3103/S1066369X23070095
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X23070095