enormalization of almost commuting pair

In this paper we give a new prove of hyperbolicity of renormalization of critical circle maps using the formalism of almost-commuting pairs. We extend renormalization to two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. Finally, we demontsrate that a two-dimensional map which lies in the stable set of the renormalization operator possesses attractor which is topologically a circle. Such a circle is <span>*critical*</span>: the dynamics on it is topologically, but not smoothly, conjugate to a rigid rotation.

<span>\*\*\*\*\*</span> L. Bers & H.L. Royden. Holomorphic families of injections. <span>*Acta Math.*</span> <span>**157**</span>(1986), 259–286.

A. de Carvalho, M. Lyubich, M. Martens. Renormalization in the H<span>é</span>non family, I: Universality but non-rigidity. <span> *Journal of Statistical Physics*</span> 121 (5-6), 611-669

W. de Melo & S. van Strien. <span>*One dimensional dynamics.*</span> Springer-Verlag, 1993.

A. Douady. Does a Julia set depend continuously on the polynomial? In <span>*Complex dynamical systems: The mathematics behind the Mandelbrot set and Julia sets*</span>. ed. R.L. Devaney, Proc. of Symposia in Applied Math., Vol 49, Amer. Math. Soc., 1994, pp. 91-138.

A. Douady, J.H. Hubbard. Etude dynamique des polynômes complexes, I-II. Pub. Math. d'Orsay, 1984.

A. Douady, J.H. Hubbard. On the dynamics of polynomial-like mappings. <span>*Ann. Sci. <span>É</span>c. Norm. Sup.*</span>, <span>**18**</span>(1985), 287-343.

E. de Faria. Proof of universality for critical circle mappings. Thesis, CUNY, 1992.

E. de Faria. Asymptotic rigidity of scaling ratios for critical circle mappings. <span>*Ergodic Theory Dynam. Systems*</span> <span>**19**</span>(1999), no. 4, 995-1035.

E. de Faria and W. de Melo. Rigidity of critical circle mappings I. <span>*J. Eur. Math. Soc. (JEMS)*</span> <span>**1**</span>(1999), no. 4, 339-392.

E. de Faria and W. de Melo. Rigidity of critical circle mappings II. <span>*J. Amer. Math. Soc.*</span>, <span>**13**</span>(2000), no. 2, 343-370.

W. de Melo, P. Guarino, Rigidity of smooth circle maps. Preprint, 2013.

A. Epstein, Towers of finite type complex analytic maps. PhD Thesis, CUNY, 1993.

A. Epstein, L. Keen, C. Tresser. The set of maps $F_{a,b}:x\mapsto x+a+\frac{b}{2\pi}\sin(2\pi x)$ with any given rotation interval is contractible. <span>*Commun. Math. Phys.*</span> <span>**173**</span>, 313-333, 1995.

A. Epstein, M. Yampolsky. The universal parabolic map. IMS at Stony Brook Preprint 2001/04.

J.-P. Eckmann and H. Epstein. On the existence of fixed points of the composition operator for circle maps. <span>*Commun. Math. Phys.*</span> <span>**107**</span> (1986), 213-231.

M. Feigenbaum, L. Kadanoff, and S. Shenker. Quasi-periodicity in dissipative systems. A renormalization group analysis. <span>*Physica*</span> <span>**5D**</span> (1982), 370-386.

D. Gaidashev, M. Yampolsky. Cylinder renormalization of Siegel disks, <span>*Exp. Math*</span>., 16(2007), 215-226

I. Gorbovickis, M. Yampolsky. Hyperbolicity of renormalization for critical circle maps with non-integer exponents, e-print ArXiv:1505.00686

J. Graczyk, D. Sands, G. <span>Ś</span>wi<span>a</span>tek, Decay of geometry for unimodal maps: negative Schwarzian case. <span>*Annals of Math.*</span>, <span>**161**</span>(2005), 613-677.

M. Herman. Conjugaison quasi-symmetrique des homeomorphismes analytiques du cercle a des rotations. Manuscript.

<span>**20**</span>(2000), 173-229.

H. Inou, M. Shishikura. The renormalization for parabolic fixed points and their perturbation. Preprint, 2008.

L. Keen. Dynamics of holomorphic self-maps of ${\Bbb C}^*$. In <span>*Holomorphic functions and moduli I*</span> (ed. D. Drasin et al.) Springer-verlag, New York, 1988.

O.E. Lanford. Renormalization group methods for critical circle mappings with general rotation number, <span>*VIIIth International Congress on Mathematical Physics (Marseille,1986)*</span>, World Sci. Publishing, Singapore, 532-536, (1987).

O.E. Lanford. Renormalization group methods for critical circle mappings. Nonlinear evolution and chaotic phenomena, <span>*NATO adv. Sci. Inst. Ser. B:Phys.*</span>,<span>**176**</span>, Plenum, New York, 25-36, (1988).

M. Lyubich. Renormalization ideas in conformal dynamics. <span>*Cambridge Seminar "Current Developments in Math.", May 1995.*</span> International Press, 1995. Cambridge, MA, 155-184.

M. Lyubich. Dynamics of quadratic polynomials, I-II. <span>*Acta Math.*</span>, <span>**178**</span>(1997), 185-297.

M. Lyubich. Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture. <span>*Ann. of Math.*</span> (2) <span>**149**</span>(1999), no. 2, 319–420.

M. Lyubich. Almost every real quadratic map is either regular or stochastic. <span>*Annals of Math.*</span>, to appear.

M. Lyubich and M.Yampolsky. Dynamics of quadratic polynomials: complex bounds for real maps. <span>*Ann. l'Inst. Fourier*</span> <span>**47**</span>, 4(1997), 1219-1255.

R.S. MacKay, I.C. Percival. Universal small-scale structure near the boundary of Siegel disks of arbitrary rotation number. <span>*Physica*</span> <span>**26D**</span>(1987), 193-202.

R. Ma<span>ñ</span><span>é</span>, P. Sad, D. Sullivan. On the dynamics of rational maps. <span>*Ann. Sci. Éc. Norm. Sup.*</span> <span>**16**</span>(1983), 193–217.

M. Martens. The periodic points of renormalization, <span>*Ann. of Math.*</span> (2) 147 (3) (1998), 435-484.

C. McMullen. Complex dynamics and renormalization. <span>*Annals of Math. Studies*</span>, v.135, Princeton Univ. Press, 1994.

C. McMullen. Renormalization and 3-manifolds which fiber over the circle. <span>*Annals of Math. Studies*</span>, Princeton University Press, 1996.

B.D. Mestel. A computer assisted proof of universality for cubic critical maps of the circle with golden mean rotation number. PhD Thesis, University of Warwick, 1985.

J. Milnor. <span>*Dynamics in one complex variable. Introductory lectures.*</span> Friedr. Vieweg & Sohn, Braunschweig, 1999.

S. Ostlund, D. Rand, J. Sethna, E. Siggia, Universal properties of the transition from quasi-periodicity to chaos in dissipative systems. <span>*Physica*</span>, <span>**8D**</span>(1983), 303-342.

E. Pujals. Questions about $C^r$ stability. Talk presented at Dynamical Systems workshop at ICTP, 2012.

D.Sullivan. Quasiconformal homeomorphisms and dynamics, topology and geometry. <span>*Proc. ICM-86, Berkeley*</span>, v. II, 1216-1228.

D.Sullivan. Bounds, quadratic differentials, and renormalization conjectures. <span>*AMS Centennial Publications. <span>**2**</span>: Mathematics into Twenty-first Century*</span> (1992).

M. Shishikura. The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. <span>*Ann. of Math.*</span> (2) <span>**147**</span>(1998), no. 2, 225-267.

G.Świactek. Rational rotation numbers for maps of the circle. <span>*Commun. Math. Phys.*</span> <span>**119**</span>(1988), 109-128.

M. Yampolsky. Complex bounds for renormalization of critical circle maps, <span>*Erg. Th. & Dyn. Systems*</span>. <span>**19**</span>(1999), 227-257.

M. Yampolsky. The attractor of renormalization and rigidity of towers of critical circle maps, <span>*Commun. Math. Phys.*</span>, <span>**218**</span>(2001), no. 3, 537-568.

M. Yampolsky. Hyperbolicity of renormalization of critical circle maps. <span>*Publ. Math. Inst. Hautes Etudes Sci.*</span> No. 96 (2002), 1–41

M. Yampolsky. Global renormalization horseshoe for critical circle maps, <span>*Commun. Math. Physics*</span>, 240(2003), 75–96. Global renormalization horseshoe.

M. Yampolsky. Siegel disks and renormalization fixed points, <span>*Fields Institute Communications*</span>, 53(2008)

J.-C. Yoccoz, Il n'ya pas de contre-example de Denjoy analytique, <span>*C.R. Acad. Sci. Paris*</span> <span>**298**</span>(1984) série I, 141-144.

{'source': 'arXiv_out/1604'}