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Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks

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Abstract

We prove the renormalization conjecture for circle diffeomorphisms with breaks, i.e., that the renormalizations of any two C 2+α-smooth (α ∈ (0, 1)) circle diffeomorphisms with a break point, with the same irrational rotation number and the same size of the break, approach each other exponentially fast in the C 2-topology. As was shown in [KKM], this result implies the following strong rigidity statement: for almost all irrational numbers ρ, any two circle diffeomorphisms with a break, with the same rotation number ρ and the same size of the break, are C 1-smoothly conjugate to each other. As we proved in [KK13], the latter claim cannot be extended to all irrational rotation numbers. These results can be considered an extension of Herman’s theory on the linearization of circle diffeomorphisms.

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References

  1. G. Arioli and H. Koch. The critical renormalization fixed point for commuting pairs of area-preserving maps. Comm. Math. Phys. 295 2 (2010), 415–429.

  2. V. I. Arnold. Small denominators I: On the mapping of a circle into itself. Izv. Akad. Nauk. Math. Serie 25 (1961), 21–86; Transl. A.M.S. Serie 2 46 (1965).

  3. V. I. Arnold. Proof of A. N. Kolmogorov’s theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian. Uspekhi Mat. Nauk SSSR 18(5) (1963), 13–40.

  4. A. Avila and R. Krikorian. Reducibility or nonuniform hyperbolicity of quasiperiodic Schrödinger cocycles. Ann. Math. 164 4 (2006), 911–940.

  5. K. Cunha and D. Smania. Renormalization for piecewise smooth homeomorphisms on the circle. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 3 (2013), 441–462.

  6. Cunha K., Smania D.: Rigidity for piecewise smooth homeomorphisms on the circle. Adv. Math. 250, 193–226 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Coullet and C. Tresser. Iteration d’endomorphismes et groupe de renormalisation. J. Phys. Colloque 39, (1978) C5–25.

  8. A. Denjoy. Sur les courbes définies par les équations differentielles à la surface du tore. J. Math. Pures Appli. Série 9 11 (1932), 333–375.

  9. Feigenbaum M. J.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  10. Feigenbaum M. J.: The universal metric properties of nonlinear transformations. J. Stat. Phys. 21, 669–706 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  11. E. de Faria and W. de Melo. Rigidity of critical circle maps I. J. Eur. Math. Soc. 1 4 (1999), 339–392.

  12. E. de Faria and W. de Melo. Rigidity of critical circle maps II. J. Am. Math. Soc. 13 2 (2000), 343–370.

  13. D. Gaidashev. Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori. Discrete Contin. Dyn. Syst. 13 1 (2005), 63–102.

  14. M. R. Herman. Sur la conjugasion differentiable des difféomorphismes du cercle a de rotations. Publ. Math. Inst. Hautes Etudes Sci. 49 (1979), 5–234.

  15. Y. Katznelson and D. Orstein. The differentiability of conjugation of certain diffeomorphisms of the circle. Ergodic Theory Dynam. Systems 9 (1989), 643–680.

  16. K. Khanin and D. Khmelev. Renormalizations and rigidity theory for circle homeomorphisms with singularities of break type. Commun. Math. Phys. 235 1 (2003), 69–124.

  17. K. Khanin and S. Kocić. Abscence of robust rigidity for circle diffeomorphisms with breaks. Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 3 (2013), 385–399.

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

  19. K. Khanin, A. Teplinsky. Robust rigidity for circle diffeomorphisms with singularities. Invent. Math. 169 (2007), 193–218.

  20. K. Khanin, A. Teplinsky. Herman’s theory revisited. Invent. Math. 178 (2009), 333–344.

  21. K. Khanin, A. Teplinsky. Renormalization horseshoe and rigidity theory for circle diffeomorphisms with breaks. Comm. Math. Phys. 320 2 (2013), 347–377.

  22. K. M. Khanin, E. B. Vul. Circle Homeomorphisms with Weak Discontinuities. Advances in Sov. Math. 3 (1991), 57–98.

  23. H. Koch and S. Kocić. Renormalization of vector fields and Diophantine invariant tori. Ergodic Theory Dynam. Systems 28 5 (2008), 1559–1585.

  24. H. Koch and S. Kocić. A renormalization group aproach to quasiperiodic motion with Brjuno frequencies. Ergodic Theory Dynam. Systems 30 (2010), 1131–1146.

  25. S. Kocić. Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori. Nonlinearity 18 (2005), 1–32.

  26. S. Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete Contin. Dyn. Syst. 29 1 (2011), 261–283.

  27. A. N. Kolmogorov. On conservation of quasiperiodic motions under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSSR 98 (1954), 527–530.

  28. M. Lyubich. Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture. Ann. Math. 149 1 (1999), 319–420.

  29. R. S. MacKay, J. D. Meiss and J. Stark. An approximate renormalization for the break-up of invariant tori with three frequencies. Physics Letters A 190 (1994), 417–424.

  30. S. Marmi, P. Moussa, and J.-C. Yoccoz. Linearization of generalized interval exchange maps. Ann. Math 176 3 (2012) 1583–1646.

  31. C. T. McMullen. Renormalization and 3-manifolds which fiber over the circle. Princeton University Press, Princeton, NJ, 1996.

  32. J. Moser. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. II 1 (1962), 1–20.

  33. S. Ostlund, D. Rand, J. Sethna, and E. Siggia, Universal properties of the transition from quasi-periodicity to chaos in dissipative systems. Physica 8D (1983), 303–342.

  34. Y. G. Sinai and K. M. Khanin. Smoothness of conjugacies of diffeomorphisms of the circle with rotations. Uspekhi Mat. Nauk 44 1 (1989), 57–82.

  35. E. C. G. Stueckelberg, and A. Petermann. La normalisation des constantes dans la theorie des quanta. Helv. Phys. Acta 26 (1953), 499–520.

  36. D. Sullivan. Bounds, quadratic differentials and renormalization conjectures. In: American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) 417–466; Amer. Math. Soc. (Providence, RI, 1992).

  37. Yampolsky M.: Hyperbolicity of renormalization of critical circle maps. Publ. Math. Inst. Hautes Etudes Sci. 96, 1–41 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yoccoz J.-C.: Conjugaison differentiable des difféomorphismes du cercle donc le nombre de rotation vérifie une condition Diophantienne. Ann. Sci. Ec. Norm. Sup. 17, 333–361 (1984)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Konstantin Khanin

  2. Department of Mathematics, University of Mississippi, University, MS, 38677-1848, USA

    Saša Kocić

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  1. Konstantin Khanin
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  2. Saša Kocić
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Correspondence to Saša Kocić.

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Khanin, K., Kocić, S. Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks. Geom. Funct. Anal. 24, 2002–2028 (2014). https://doi.org/10.1007/s00039-014-0309-0

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