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A circle diffeomorphism with breaks that is absolutely continuously linearizable

Published online by Cambridge University Press:  04 July 2016

ALEXEY TEPLINSKY
ALEXEY TEPLINSKY*
Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska Street, Kiev 01601, Ukraine email [email protected], [email protected]
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Abstract

In this paper we answer positively to a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case where its breaks are lying on pairwise distinct trajectories. An example constructed is a piecewise linear circle homeomorphism that has four break points lying on distinct trajectories and whose invariant measure is absolutely continuous with respect to the Lebesgue measure. The irrational rotation number for our example can be chosen to be a Roth number, but not of bounded type.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 371 - 383
Copyright
© Cambridge University Press, 2016 

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References

Adouani, A. and Marzougui, H.. Singular measures for class P-circle homeomorphisms with several break points. Ergod. Th. & Dynam. Sys. 34 (2014), 423–456.Google Scholar
Adouani, A.. Conjugation between circle maps with severeal break points. Ergod. Th. & Dynam. Sys., to appear, doi:10.1017/etds.2015.32.Google Scholar
Cunha, K. and Smania, D.. Rigidity for piecewise smooth homeomorphisms on the circle. Adv. Math. 250 (2014), 193–226.CrossRefGoogle Scholar
Dzhalilov, A. A. and Khanin, K. M.. On an invariant measure for homeomorphisms of a circle with a point of break. Funct. Anal. Appl. 32 (1998), 153–161.CrossRefGoogle Scholar
Dzhalilov, A. and Liousse, I.. Circle homeomorphisms with two break points. Nonlinearity 19 (2006), 1951–1968.Google Scholar
Dzhalilov, A. A., Mayer, D. and Safarov, U. A.. Piecewise-smooth circle homeomorphisms with several break points. Izv. Math. 76 (2012), 94–112.CrossRefGoogle Scholar
Dzhalilov, A., Mayer, D. and Safarov, U.. Conjugacies between P-homeomorphisms with several breaks. Preprint, 2014, arXiv:1408.5732 [math.DS], http://iopscience.iop.org/article/10.1088/0951-7715/28/7/2441.Google Scholar
Herman, M.-R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5–233.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1997.Google Scholar
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
Khanin, K. and Teplinsky, A.. Herman’s theory revisited. Invent. Math. 178 (2009), 333–344.Google Scholar
Khanin, K. and Teplinsky, A.. Renormalization horseshoe and rigidity for circle diffeomorphisms with breaks. Comm. Math. Phys. 320 (2013), 347–377.CrossRefGoogle Scholar