Rigidity of critical circle maps. (English) Zbl 1400.37047

Guarino, Pablo; Martens, Marco; De Melo, Welington

Duke Math. J. 167, No. 11, 2125-2188 (2018).

A critical circle map is an orientation-preserving \(C^4\) circle homeomorphism \(f\) with exactly one critical point \(c\) such that, in a neighborhood of \(c\), the map \(f\) can be written as \(f(t)=f(c)+(\phi (t))^{2d+1}\) where \(\phi\) is a \(C^4\) local diffeomorphism with \(\phi (c)=0\) and \(d \in \mathbb{N}\) with \(d\geq 1\). The criticality (also called order, type or exponent ) of the critical point \(c\) is the odd integer \(2d+1\). The critical point is also called nonflat.
A \(C^4\) critical circle map \(f\) with irrational rotation number generates a sequence \(\{\mathcal{R}^n(f)\}_{n \in \mathbb{N}}\) of commuting pairs of interval maps, each such map being the renormalization of the previous one. The authors observe that a critical commuting pair is a special case of a generalized interval exchange map of two intervals and the renormalization operator is the restriction of the Zorich accelerated version of the Rauzy-Veech renormalization for interval exchange maps.
The main result in the paper states that there exists a universal constant \(\lambda\), a point in the open interval \((0,\,1),\) such that given two \(C^4\) critical circle maps \(f\) and \(g\) with the same irrational rotation number and the same criticality, there exists \(C >0\) (a constant that depends on \(f\) and \(g\)) such that for all \(n \in \mathbb{N}\) we have \( d_2 (\mathcal{R}^n(f), \mathcal{R}^n(g)) \leq C \lambda^n,\) where \(d_2\) denotes the \(C^2\)-distance in the space of \(C^2\) critical commuting pairs.
This theorem implies the following rigidity theorem: given two \(C^4\) circle homeomorphisms \(f\) and \(g\) with the same irrational rotation number and with a unique critical point of the same odd type. If \(h\) denotes the unique topological conjugacy between \(f\) and \(g\) that maps the critical point of \(f\) to the critical point of \(g\), then the following three statements hold: (1) \(h\) is a \(C^1\)-diffeomorphism; (2) \(h\) is a \(C^{1+\alpha}\) at the critical point of \(f\) for a universal \(\alpha >0\); and (3) for a full Lebesgue measure set of rotation numbers, \(h\) is a \(C^{1+\alpha}\)-diffeomorphism.
This paper has 13 sections, an appendix and a list of references.

Reviewer: Fernanda Botelho (Memphis)

MSC:

37E10	Dynamical systems involving maps of the circle
37E05	Dynamical systems involving maps of the interval
37E20	Universality and renormalization of dynamical systems
37E45	Rotation numbers and vectors

Keywords:

critical circle map; smooth rigidity; renormalization

References:

[1]	A. Ávila, On rigidity of critical circle maps, Bull. Braz. Math. Soc. (N.S.) 44 (2013), 611–619. · Zbl 1314.37028
[2]	E. de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergodic Theory Dynam. Systems 19 (1999), 995–1035. · Zbl 0996.37045
[3]	E. de Faria and W. de Melo, Rigidity of critical circle mappings, I, J. Eur. Math. Soc. (JEMS) 1 (1999), 339–392. · Zbl 0988.37047
[4]	E. de Faria and W. de Melo, Rigidity of critical circle mappings, II, J. Amer. Math. Soc. 13 (2000), 343–370. · Zbl 0988.37048
[5]	E. de Faria and P. Guarino, Real bounds and Lyapunov exponents, Discrete Contin. Dyn. Syst. 36 (2016), 1957–1982. · Zbl 1364.37091
[6]	W. de Melo, “Rigidity and renormalization in one-dimensional dynamical systems” in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, 765–778. · Zbl 0914.58016
[7]	W. de Melo and A. A. Pinto, Rigidity of \(C^{2}\) infinitely renormalizable unimodal maps, Comm. Math. Phys. 208 (1999), 91–105. · Zbl 0982.37026
[8]	W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. (3) 25, Springer, Berlin, 1993. · Zbl 0791.58003
[9]	M. J. Feigenbaum, L. P. Kadanoff, and S. S. Shenker, Quasiperiodicity in dissipative systems: A renormalization group analysis, Phys. D 5 (1982), 370–386.
[10]	J. Graczyk and G. Świa̧tek, Critical circle maps near bifurcation, Comm. Math. Phys. 176 (1996), 227–260. · Zbl 0857.58034
[11]	P. Guarino and W. de Melo, Rigidity of smooth critical circle maps, J. Eur. Math. Soc. (JEMS) 19 (2017), 1729–1783. · Zbl 1366.37098
[12]	M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5–233.
[13]	M.-R. Herman, Conjugaison quasi-symétrique des homéomorphismes du cercle à des rotations, preprint, http://www.math.kyoto-u.ac.jp/mitsu/Herman/qsconj1/index.html (accessed 6 June 2018).
[14]	K. Khanin and A. Teplinsky, Robust rigidity for circle diffeomorphisms with singularities, Invent. Math. 169 (2007), 193–218. · Zbl 1151.37035
[15]	D. Khmelev and M. Yampolsky, The rigidity problem for analytic critical circle maps, Mosc. Math. J. 6 (2006), 317–351. · Zbl 1124.37024
[16]	O. E. Lanford, III, “Renormalization group methods for critical circle mappings with general rotation number” in VIIIth International Congress on Mathematical Physics (Marseille, 1986), World Scientific, Singapore, 1987, 532–536.
[17]	O. E. Lanford, III, “Renormalization group methods for circle mappings” in Nonlinear Evolution and Chaotic Phenomena (Noto, 1987), NATO Adv. Sci. Inst. Ser. B Phys. 176, Plenum, New York, 1988, 25–36.
[18]	M. Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Ann. of Math. (2) 149 (1999), 319–420. · Zbl 0945.37012
[19]	M. Martens, The periodic points of renormalization, Ann. of Math. (2) 147 (1998), 543–584. · Zbl 0936.37017
[20]	C. T. McMullen, Complex Dynamics and Renormalization, Ann. of Math. Stud. 135, Princeton Univ. Press, Princeton, 1994. · Zbl 0822.30002
[21]	C. T. McMullen, “Rigidity and inflexibility in conformal dynamics” in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, 841–855. · Zbl 0914.58015
[22]	S. Östlund, D. Rand, J. Sethna, and E. Siggia, Universal properties of the transition from quasiperiodicity to chaos in dissipative systems, Phys. D 8 (1983), 303–342. · Zbl 0538.58025
[23]	D. Rand, Global phase space universality, smooth conjugacies and renormalisation, I: The \(C^{1+α}\) case, Nonlinearity 1 (1988), 181–202. · Zbl 0697.58029
[24]	D. Rand, “Universality and renormalisation in dynamical systems” in New Directions in Dynamical Systems, London Math. Soc. Lecture Note Ser. 127, Cambridge Univ. Press, Cambridge, 1988, 1–56. · Zbl 0651.58022
[25]	D. Rand, Existence, nonexistence and universal breakdown of dissipative golden invariant tori, I: Golden critical circle maps, Nonlinearity 5 (1992), 639–662. · Zbl 0759.58040
[26]	S. J. Shenker, Scaling behavior in a map of a circle onto itself: Empirical results, Phys. D 5 (1982), 405–411.
[27]	G. Świa̧tek, Rational rotation numbers for maps of the circle, Comm. Math. Phys. 119 (1988), 109–128.
[28]	M. Yampolsky, Complex bounds for renormalization of critical circle maps, Ergodic Theory Dynam. Systems 19 (1999), 227–257. · Zbl 0918.58049
[29]	M. Yampolsky, The attractor of renormalization and rigidity of towers of critical circle maps, Comm. Math. Phys. 218 (2001), 537–568. · Zbl 0978.37033
[30]	M. Yampolsky, Hyperbolicity of renormalization of critical circle maps, Publ. Math. Inst. Hautes Études Sci. 96 (2002), 1–41. · Zbl 1030.37027
[31]	M. Yampolsky, Renormalization horseshoe for critical circle maps, Comm. Math. Phys. 240 (2003), 75–96. · Zbl 1078.37034
[32]	J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. Éc. Norm. Supér. (4) 17 (1984), 333–359.
[33]	J.-C. Yoccoz, Il n’y a pas de contre-exemple de Denjoy analytique, C. R. Math. Acad. Sci. Paris 298 (1984), 141–144. · Zbl 0573.58023