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 Uspekhi Mat. Nauk, 2004, Volume 59, Issue 2(356), Pages 137–160 (Mi umn722)

Rigidity for circle diffeomorphisms with singularities

A. Yu. Teplinskiia, K. M. Khaninbcd

a Institute of Mathematics, Ukrainian National Academy of Sciences
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
c Heriot Watt University
d Isaac Newton Institute for Mathematical Sciences

Abstract: This paper reviews recent results related to rigidity theory for circle diffeomorphisms with singularities. Both diffeomorphisms with a break point (sometimes called a ‘fracture-type singularity’ or ‘weak discontinuity’) and critical circle maps are discussed. In the case of breaks, results are presented on the global hyperbolicity of the renormalization operator; this property implies the existence of an attractor of the Smale horseshoe type. It is also shown that for maps with singularities rigidity is stronger than for diffeomorphisms, in the sense that rigidity is not violated for non-generic rotation numbers, which are abnormally well approximable by rationals. In the case of critical rotations of the circle it is proved that any two such rotations with the same order of the singular point and the same irrational rotation number are $C^1$-smoothly conjugate.

DOI: https://doi.org/10.4213/rm722

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English version:
Russian Mathematical Surveys, 2004, 59:2, 329–353

Bibliographic databases:

UDC: 517.9
MSC: Primary 37E10; Secondary 37E20, 37E45, 37J40, 11J70, 11J25

Citation: A. Yu. Teplinskii, K. M. Khanin, “Rigidity for circle diffeomorphisms with singularities”, Uspekhi Mat. Nauk, 59:2(356) (2004), 137–160; Russian Math. Surveys, 59:2 (2004), 329–353

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Valdez R., “Self-similarity of the mandelbrot set for real essentially bounded combinatorics”, Discrete Contin. Dyn. Syst., 16:4 (2006), 897–922
2. D. V. Khmelev, M. Yampolsky, “The rigidity problem for analytic critical circle maps”, Mosc. Math. J., 6:2 (2006), 317–351
3. Diaz-Espinosa O., de la Llave R., “Renormalization and central limit theorem for critical dynamical systems with weak external noise”, J. Mod. Dyn., 1:3 (2007), 477–543
4. Khanin K., Teplinsky A., “Robust rigidity for circle diffeomorphisms with singularities”, Invent. Math., 169:1 (2007), 193–218
5. Teplins'kyi O.Yu., “Hyperbolic horseshoe for circle diffeomorphisms with a break”, Nonlinear Oscil. (N. Y.), 11:1 (2008), 114–134
6. Dzhalilov A., Akin H., Temir S., “Conjugations between circle maps with a single break point”, J. Math. Anal. Appl., 366:1 (2010), 1–10
7. Teplins'kyi O.Yu., Khanin K.M., “Smooth Conjugacy of Circle Diffeomorphisms With Break”, Nonlinear Oscillations, 13:1 (2010), 112–127
8. Teplins'kyi O.Yu., “EXAMPLES OF C-1-SMOOTHLY CONJUGATE DIFFEOMORPHISMS OF THE CIRCLE WITH BREAK THAT ARE NOT C1+gamma-SMOOTHLY CONJUGATE”, Ukrainian Math J, 62:8 (2011), 1267–1284
9. A. A. Dzhalilov, D. Mayer, U. A. Safarov, “Piecewise-smooth circle homeomorphisms with several break points”, Izv. Math., 76:1 (2012), 94–112
10. HABIBULLA AKHADKULOV, AKHTAM DZHALILOV, DIETER MAYER, “On conjugations of circle homeomorphisms with two break points”, Ergod. Th. Dynam. Sys, 2012, 1
11. Habibulla Akhadkulov, Mohd Salmi Md Noorani, “On Absolute Continuity of Conjugations between Circle Maps with Break Points”, Abstract and Applied Analysis, 2012 (2012), 1
12. Konstantin Khanin, Alexey Teplinsky, “Renormalization Horseshoe and Rigidity for Circle Diffeomorphisms with Breaks”, Commun. Math. Phys, 2013
13. Akhatkulov S. Noorani Mohd Salmi Md Akhadkulov H., “On Conjugacies of Circle Maps With Singular Points”, Statistics and Operational Research International Conference, AIP Conference Proceedings, 1613, ed. Shitan M. Lee L. Eshkuvatov Z., Amer Inst Physics, 2014, 266–274
14. Akhadkulov H. Noorani Mohd Salmi Md Akhatkulov S., “On Smoothness of Conjugations Between Break Equivalent Circle Maps”, Proceedings of the 3rd International Conference on Mathematical Sciences, AIP Conference Proceedings, 1602, ed. Zin W. DzulKifli S. Razak F. Ishak A., Amer Inst Physics, 2014, 749–753
15. Hongfei Cui, Yiming Ding, “Renormalization and conjugacy of piecewise linear Lorenz maps”, Advances in Mathematics, 271 (2015), 235
16. Akhadkulov H., Noorani Mohd Salmi Md, “On Conjugations of P-Homeomorphisms”, Indag. Math.-New Ser., 26:1 (2015), 280–292
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