Absence of robust rigidity for circle maps with breaks
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, pp. 385-399.

We give examples of analytic circle maps with singularities of break type with the same rotation number and the same size of the break for which no conjugacy is Lipschitz continuous. In the second part of the paper, we discuss a class of rotation numbers for which a conjugacy is C 1 -smooth, although the numbers can be strongly non-Diophantine (Liouville). For the rotation numbers in this class, we construct examples of analytic circle maps with breaks, for which the conjugacy is not C 1+α smooth, for any α>0.

Nous donnons des exemples dʼapplications du cercle analytiques avec des singularités de type rupture avec le même nombre de rotation et la même taille de rupture pour lesquelles aucune conjugaison nʼest lipschitzienne. Dans la deuxième partie de lʼarticle, nous étudions une classe de nombres de rotation pour lesquels il y a une conjugaison de classe C 1 , alors même que les nombres de rotation peuvent être fortement non-diophantiens (Liouville). Pour les nombres de rotation de cette classe, nous construisons des exemples dʼapplications du cercle analytiques avec des singularités de type rupture, pour lesquelles la conjugaison nʼest de classe C 1+α pour aucun α>0.

@article{AIHPC_2013__30_3_385_0,
     author = {Khanin, Konstantin and Koci\'c, Sa\v{s}a},
     title = {Absence of robust rigidity for circle maps with breaks},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {385--399},
     publisher = {Elsevier},
     volume = {30},
     number = {3},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.08.004},
     mrnumber = {3061428},
     zbl = {06295425},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.08.004/}
}
TY  - JOUR
AU  - Khanin, Konstantin
AU  - Kocić, Saša
TI  - Absence of robust rigidity for circle maps with breaks
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 385
EP  - 399
VL  - 30
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.08.004/
DO  - 10.1016/j.anihpc.2012.08.004
LA  - en
ID  - AIHPC_2013__30_3_385_0
ER  - 
%0 Journal Article
%A Khanin, Konstantin
%A Kocić, Saša
%T Absence of robust rigidity for circle maps with breaks
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 385-399
%V 30
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.08.004/
%R 10.1016/j.anihpc.2012.08.004
%G en
%F AIHPC_2013__30_3_385_0
Khanin, Konstantin; Kocić, Saša. Absence of robust rigidity for circle maps with breaks. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, pp. 385-399. doi : 10.1016/j.anihpc.2012.08.004. http://www.numdam.org/articles/10.1016/j.anihpc.2012.08.004/

[1] M.R. Herman, Sur la conjugasion différentiable des difféomorphismes du cercle à des rotations, Publ. Math. Inst. Hautes Etudes Sci. 49 (1979), 5-234 | EuDML | Numdam | MR

[2] J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition Diophantienne, Ann. Sci. Éc. Norm. Supér. 17 (1984), 333-361 | EuDML | Numdam | MR | Zbl

[3] Y. Katznelson, D. Orstein, The differentiability of conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dynam. Systems 9 (1989), 643-680 | MR | Zbl

[4] Ya.G. Sinai, K.M. Khanin, Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Uspekhi Mat. Nauk 44 no. 1 (1989), 57-82 | MR | Zbl

[5] K. Khanin, A. Teplinsky, Hermanʼs theory revisited, Invent. Math. 178 (2009), 333-344 | MR | Zbl

[6] V.I. ArnolʼD, Small denominators I: On the mapping of a circle into itself, Izv. Akad. Nauk Mat. Ser. 25 (1961), 21-86, Amer. Math. Soc. Transl. Ser. 2 46 (1965) | MR

[7] K. Khanin, A. Teplinsky, Robust rigidity for circle diffeomorphisms with singularities, Invent. Math. 169 (2007), 193-218 | MR | Zbl

[8] K. Khanin, D. Khmelev, Renormalizations and rigidity theory for circle homeomorphisms with singularities of break type, Comm. Math. Phys. 235 no. 1 (2003), 69-124 | MR | Zbl

[9] S. Marmi, P. Moussa, J.-C. Yoccoz, Linearization of generalized interval exchange maps, arXiv:1003.1191 (2010) | MR | Zbl

[10] A. Denjoy, Sur les courbes définies par les équations differentielles à la surface du tore, J. Math. Pures Appl. (9) 11 (1932), 333-375 | EuDML | JFM | Numdam

[11] J.-C. Yoccoz, Il nʼy a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris Sér. I. Math. 298 no. 7 (1984), 141-144 | MR | Zbl

[12] E. De Faria, W. De Melo, Rigidity of critical circle maps I, J. Eur. Math. Soc. 1 no. 4 (1999), 339-392 | EuDML | MR | Zbl

[13] E. De Faria, W. De Melo, Rigidity of critical circle maps II, J. Amer. Math. Soc. 13 no. 2 (2000), 343-370 | MR | Zbl

[14] M. Yampolsky, Hyperbolicity of renormalization of critical circle maps, Publ. Math. Inst. Hautes Etudes Sci. 96 (2002), 1-41 | EuDML | Numdam | MR | Zbl

[15] A. Avila, On rigidity of critical circle maps, preprint Univ. Paris 6, 2005. | MR

[16] K. Khanin, A. Teplinsky, Renormalization horseshoe and rigidity theory for circle diffeomorphisms with breaks, preprint IML-0910s-17, 2010. | MR

[17] K. Khanin, S. Kocić, Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks, preprint mp-arc 12-38, 2012. | MR

[18] K. Khanin, S. Kocić, E. Mazzeo, C 1 -rigidity of circle diffeomorphisms with breaks for almost all rotation numbers, preprint mp-arc 11-102, 2011.

[19] A. Dzhalilov, A. Teplinsky, Certain examples of circle diffeomorphisms with a break, Dopov. NAN Ukr. 9 (2010), 18-23 | Zbl

[20] A. Teplinsky, Examples of circle diffeomorphisms with a break which are C 1 -smoothly but not C 1+γ -smoothly conjugate, Ukrainian Math. J. 62 no. 8 (2011), 1267-1284, http://dx.doi.org/10.1007/s11253-011-0428-9

[21] K.M. Khanin, E.B. Vul, Circle homeomorphisms with weak discontinuities, Adv. Soviet. Math. 3 (1991), 57-98 | MR | Zbl

Cited by Sources: