We prove the existence of two real-analytic diffeomorphisms of the circle with break of the same size and an irrational rotation number of semibounded type that are not C 1+γ-smoothly conjugate for any γ > 0. In this way, we show that the previous result concerning the C 1-smoothness of conjugacy for these mappings is the exact estimate of smoothness for this conjugacy.
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References
A. Denjoy, “Sur les courbes définies par les equation différentielles a la surface du tore,” J. Math. Pure. Appl., 11, 333–375 (1932).
V. I. Arnol’d, “Small denominators. I. On the maps of a circle onto itself,” Izv. Akad. Nauk SSSR, 25, No. 1, 21–86 (1961).
M.-R. Herman, “Sur la conjugaison differentiable des diffeomorphismes du cercle a des rotations,” I.H.E.S. Publ. Math., 49, 5–233 (1979).
A. Yu. Teplinskii and K. M. Khanin, “Rigidity for diffeomorphisms of the circle with singularities,” Usp. Mat. Nauk, 59, No. 2, 137–160 (2004).
K. Khanin and A. Teplinsky, “Robust rigidity for circle diffeomorphisms with singularities,” Invent. Math., 169, No. 1, 193–218 (2007).
O. Yu. Teplins’kyi and K. M. Khanin, “Smooth conjugation of diffeomorphisms of the circle with break,” Nelin. Kolyvannya, 13, No. 1, 100–114 (2010).
A. Avila, On Rigidity of Critical Circle Maps, Preprint, Univ. Paris 6, Paris (2005); Available from http://www.impa.br/~avila/circle.pdf.
E. V. Vul and K. M. Khanin, ‘Homeomorphisms of the circle with singularities of break type,” Usp. Mat. Nauk, 45, No. 3, 189–190 (1990).
K. M. Khanin and E. B. Vul, “Circle homeomorphisms with weak discontinuities,” in: Proceedings of the International Conference “Dynamical Systems and Statistical Mechanics” (Moscow, 1991), American Mathematical Society, Providence, RI (1991), pp. 57–98.
K. Khanin and D. Khmelev, “Renormalizations and rigidity theory for circle homeomorphisms with singularities of break type,” Commun. Math. Phys., 235, No. 1, 69–124 (2003).
O. Yu. Teplins’kyi, “Hyperbolic horseshoe for diffeomorphisms of the circle with break,” Nelin. Kolyvannya, 11, No. 1, 112–127 (2008).
I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).
A. Ya. Khinchin, Continued Fractions [in Russian], Fizmatgiz, Moscow (1960).
Y. Pomeau and P. Manneville, “Intermittent transition to turbulence in dissipative dynamical systems,” Commun. Math. Phys., 74, No. 2, 189–197 (1980).
J. Milnor, Dynamics in One Complex Variable, Princeton University, Princeton (2006).
P. M. Bleher and M.V. Jakobson, “Absolutely continuous invariant measures for some maps of the circle,” Stat. Phys. Dinam. Syst. Progr. Phys., 10, 303–315 (1985).
N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland, Amsterdam (1958).
W. Rudin, Functional Analysis, McGraw-Hill, New York (1973).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 8, pp. 1092–1105, August, 2010.
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Teplins’kyi, O.Y. Examples of C 1-smoothly conjugate diffeomorphisms of the circle with break that are not C 1+γ-smoothly conjugate. Ukr Math J 62, 1267–1284 (2011). https://doi.org/10.1007/s11253-011-0428-9
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DOI: https://doi.org/10.1007/s11253-011-0428-9