Real bounds and quasisymmetric rigidity of multicritical circle maps
HTML articles powered by AMS MathViewer
- by Gabriela Estevez and Edson de Faria PDF
- Trans. Amer. Math. Soc. 370 (2018), 5583-5616 Request permission
Abstract:
Let $f, g:S^1\to S^1$ be two $C^3$ critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if $h:S^1\to S^1$ is a topological conjugacy between $f$ and $g$ and $h$ maps the critical points of $f$ to the critical points of $g$, then $h$ is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T. Clark and S. van Strien preprint, 2014. However, unlike their proof, which relies on heavy complex-analytic machinery, our proof uses purely real-variable methods and is valid for non-integer critical exponents as well. We do not require $h$ to preserve the power-law exponents at corresponding critical points.References
- Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
- Artur Avila, On rigidity of critical circle maps, Bull. Braz. Math. Soc. (N.S.) 44 (2013), no. 4, 611–619. MR 3167125, DOI 10.1007/s00574-013-0027-5
- Artur Avila and Mikhail Lyubich, The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes, Publ. Math. Inst. Hautes Études Sci. 114 (2011), 171–223. MR 2854860, DOI 10.1007/s10240-011-0034-2
- Lennart Carleson, On mappings, conformal at the boundary, J. Analyse Math. 19 (1967), 1–13. MR 215986, DOI 10.1007/BF02788706
- T. Clark and S. van Strien, Quasisymmetric rigidity in one-dimensional dynamics, preprint, 2014.
- G. Estevez, E. de Faria, and P. Guarino, Beau bounds for multicritical circle maps, Indagationes Mathematicae (to appear), DOI 10.1016/j.indag.2017.12.007, arXiv:1611.00722
- Edson de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergodic Theory Dynam. Systems 19 (1999), no. 4, 995–1035. MR 1709428, DOI 10.1017/S0143385799133959
- Edson de Faria and Welington de Melo, Rigidity of critical circle mappings. I, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 4, 339–392. MR 1728375, DOI 10.1007/s100970050011
- Edson de Faria and Welington de Melo, Rigidity of critical circle mappings. II, J. Amer. Math. Soc. 13 (2000), no. 2, 343–370. MR 1711394, DOI 10.1090/S0894-0347-99-00324-0
- Edson de Faria, Welington de Melo, and Alberto Pinto, Global hyperbolicity of renormalization for $C^r$ unimodal mappings, Ann. of Math. (2) 164 (2006), no. 3, 731–824. MR 2259245, DOI 10.4007/annals.2006.164.731
- Edson de Faria and Pablo Guarino, Real bounds and Lyapunov exponents, Discrete Contin. Dyn. Syst. 36 (2016), no. 4, 1957–1982. MR 3411549, DOI 10.3934/dcds.2016.36.1957
- P. Guarino, Rigidity conjecture for $C^3$ critical circle maps, thesis, IMPA, 2013.
- Pablo Guarino and Welington de Melo, Rigidity of smooth critical circle maps, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 6, 1729–1783. MR 3646874, DOI 10.4171/JEMS/704
- John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582, DOI 10.1007/BF02684769
- M. Herman, Conjugaison quasi-simétrique des homéomorphismes du cercle à des rotations (manuscript), 1988. (See also the translation by A. Chéritat, Quasisymmetric conjugacy of analytic circle homeomorphisms to rotations, http://www. math.univ-toulouse.fr/$\sim$cheritat/.)
- K. Khanin and A. Teplinsky, Robust rigidity for circle diffeomorphisms with singularities, Invent. Math. 169 (2007), no. 1, 193–218. MR 2308853, DOI 10.1007/s00222-007-0047-0
- D. Khmelev and M. Yampolsky, The rigidity problem for analytic critical circle maps, Mosc. Math. J. 6 (2006), no. 2, 317–351, 407 (English, with English and Russian summaries). MR 2270617, DOI 10.17323/1609-4514-2006-6-2-317-351
- Mikhail Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Ann. of Math. (2) 149 (1999), no. 2, 319–420. MR 1689333, DOI 10.2307/120968
- Mikhail Lyubich, Almost every real quadratic map is either regular or stochastic, Ann. of Math. (2) 156 (2002), no. 1, 1–78. MR 1935840, DOI 10.2307/3597183
- Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347, DOI 10.1515/9781400865178
- Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
- L. Palmisano, Quasi-symmetric conjugacy for circle maps with a flat interval. Preprint, http://arxiv.org/abs/1510.01703, 2015.
- Carsten Lunde Petersen, The Herman-Światek theorems with applications, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 211–225. MR 1765090
- Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
- Grzegorz Świątek, Rational rotation numbers for maps of the circle, Comm. Math. Phys. 119 (1988), no. 1, 109–128. MR 968483, DOI 10.1007/BF01218263
- Michael Yampolsky, The attractor of renormalization and rigidity of towers of critical circle maps, Comm. Math. Phys. 218 (2001), no. 3, 537–568. MR 1828852, DOI 10.1007/PL00005561
- Michael Yampolsky, Hyperbolicity of renormalization of critical circle maps, Publ. Math. Inst. Hautes Études Sci. 96 (2002), 1–41 (2003). MR 1985030, DOI 10.1007/s10240-003-0007-1
- Michael Yampolsky, Renormalization horseshoe for critical circle maps, Comm. Math. Phys. 240 (2003), no. 1-2, 75–96. MR 2004980, DOI 10.1007/s00220-003-0891-8
- J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 333–359 (French, with English summary). MR 777374, DOI 10.24033/asens.1475
- J.-C. Yoccoz, Structure des orbites des homéomorphismes analytiques posedant un point critique (manuscript), 1989.
- Jean-Christophe Yoccoz, Il n’y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 7, 141–144 (French, with English summary). MR 741080
Additional Information
- Gabriela Estevez
- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo SP, Brasil
- Email: gestevez@ime.usp.br
- Edson de Faria
- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo SP, Brasil
- MR Author ID: 357550
- Email: edson@ime.usp.br
- Received by editor(s): December 23, 2015
- Received by editor(s) in revised form: December 9, 2016
- Published electronically: February 19, 2018
- Additional Notes: This work was supported by “Projeto Temático Dinâmica em Baixas Dimensões” FAPESP Grant 2011/16265-2 and by CAPES (PROEX)
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 5583-5616
- MSC (2010): Primary 37E10; Secondary 37E20, 37F10, 37A05, 37C15
- DOI: https://doi.org/10.1090/tran/7177
- MathSciNet review: 3812112