Absence of robust rigidity for circle maps with breaks
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, p. 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 sa},
     title = {Absence of robust rigidity for circle maps with breaks},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {30},
     number = {3},
     year = {2013},
     pages = {385-399},
     doi = {10.1016/j.anihpc.2012.08.004},
     zbl = {06295425},
     mrnumber = {3061428},
     language = {en},
     url = {http://www.numdam.org/item/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/item/AIHPC_2013__30_3_385_0/

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