Dynamics of multicritical circle maps. (English) Zbl 1506.37053
Summary: This paper presents a survey of recent and not so recent results concerning the study of smooth homeomorphisms of the circle with a finite number of non-flat critical points, an important topic in the area of One-dimensional Dynamics. We focus on the analysis of the fine geometric structure of orbits of such dynamical systems, as well as on certain ergodic-theoretic and complex-analytic aspects of the subject. Finally, we review some conjectures and open questions in this field.
MSC:
| 37E10 | Dynamical systems involving maps of the circle |
| 37E20 | Universality and renormalization of dynamical systems |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
References:
| [1] | Ahlfors, L.V.: Lectures on quasi-conformal mappings. In: University Lecture Series, vol. 30, 2nd edn. American Mathematical Society (2006) · Zbl 1103.30001 |
| [2] | Arnol’d, V.I.: Small denominators I. Mappings of the circle onto itself. Izv. Akad. Nauk. Math. Ser. 25, 21-86 (1961) (Translations of the Amer. Math. Soc. (series 2)46 (1965), 213-284) · Zbl 0152.41905 |
| [3] | Avila, A.: Dynamics of renormalization operators. In: Proceedings of the International Congress of Mathematicians. Hyderabad, India (2010) · Zbl 1252.37030 |
| [4] | Avila, A., On rigidity of critical circle maps, Bull. Braz. Math. Soc., 44, 611-619 (2013) · Zbl 1314.37028 |
| [5] | Avila, A.; Kocsard, A., Cohomological equations and invariant distributions for minimal circle diffeomorphisms, Duke Math. J., 158, 501-536 (2011) · Zbl 1225.37052 |
| [6] | Avila, A.; Lyubich, M., The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes, Publ. Math. IHES, 114, 171-223 (2011) · Zbl 1286.37047 |
| [7] | Bohl, P., Uber die hinsichtlich der unabhängigen variabeln periodische, Acta Math., 40, 321-336 (1916) · JFM 46.0682.02 |
| [8] | Bordignon, L.; Iglesias, J.; Portela, A., About \(C^1\)-minimality of the hyperbolic Cantor sets, Bull. Braz. Math. Soc., 45, 525-542 (2014) · Zbl 1321.37021 |
| [9] | Boyland, P., Bifurcations of circle maps: Arnol’d tongues, bistability and rotation intervals, Commun. Math. Phys., 106, 353-381 (1986) · Zbl 0612.58032 |
| [10] | Carleson, L.; Gamelin, TW, Complex Dynamics (1993), New York: Springer-Verlag, New York · Zbl 0782.30022 |
| [11] | Chenciner, A.; Gambaudo, J-M; Tresser, C., Une remarque sur la structure des endomorphismes de degré \(1\) du cercle, C. R. Acad. Sci. Paris. Série I, 299, 145-148 (1984) · Zbl 0584.58004 |
| [12] | Clark, T.; Trejo, S., The boundary of chaos for interval mappings, Proc. London Math. Soc., 121, 1427-1467 (2020) · Zbl 1468.37040 |
| [13] | Clark, T.; van Strien, S.; Trejo, S., Complex bounds for real maps, Comm. Math. Phys., 355, 1001-1119 (2017) · Zbl 1379.37086 |
| [14] | Crovisier, S.; Guarino, P.; Palmisano, L., Ergodic properties of bimodal circle maps, Ergod. Theory Dyn. Syst., 39, 1462-1500 (2019) · Zbl 1415.37008 |
| [15] | Cvitanović, P.; Gunaratne, GH; Vinson, MJ, On the mode-locking universality for critical circle maps, Nonlinearity, 3, 873-885 (1990) · Zbl 0712.58042 |
| [16] | Cvitanović, P.; Shraiman, B.; Söderberg, B., Scaling laws for mode locking in circle maps, Physica Scrypta, 32, 263-270 (1985) · Zbl 1063.37524 |
| [17] | Davie, A., Period doubling for \(C^{2+\varepsilon }\) mappings, Commun. Math. Phys., 176, 261-272 (1996) · Zbl 0906.58012 |
| [18] | Denjoy, A., Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pure Appl., 11, 333-375 (1932) · Zbl 0006.30501 |
| [19] | Dixon, TW; Gherghetta, T.; Kenny, BG, Universality in the quasiperiodic route to chaos, Chaos, 6, 32-42 (1996) · Zbl 1055.37517 |
| [20] | Douady, A.: Disques de Siegel et anneaux de Herman. In: Sém. Bourbaki 1986/87, Astérisque, 1986/87, pp. 151-172 (1987) · Zbl 0638.58023 |
| [21] | Douady, A., Does a Julia set depend continuously on the polynomial? Complex dynamical systems, Proc. Symp. Appl. Math., 49, 91-138 (1994) · Zbl 0934.30023 |
| [22] | Douady, A.; Hubbard, J., On the dynamics of polynomial-like mappings, Ann. Sci. Ec. Norm. Sup., 18, 287-343 (1985) · Zbl 0587.30028 |
| [23] | Eliasson, H., Fayad, B., Krikorian, R.: Jean-Christophe Yoccoz and the theory of circle diffeomorphisms. Available at arXiv:1810.07107 · Zbl 1454.37040 |
| [24] | Epstein, A.; Keen, L.; Tresser, C., 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, Commun. Math. Phys., 173, 313-333 (1995) · Zbl 0839.58021 |
| [25] | Estevez, G.; de Faria, E., Real bounds and quasisymmetric rigidity of multicritical circle maps, Trans. Amer. Math. Soc., 370, 5583-5616 (2018) · Zbl 1388.37048 |
| [26] | Estevez, G.; de Faria, E.; Guarino, P., Beau bounds for multicritical circle maps, Indagationes Mathematicæ, 29, 842-859 (2018) · Zbl 1394.37071 |
| [27] | Estevez, G., Guarino, P.: Renormalization of multicritical circle maps (submitted) · Zbl 1394.37071 |
| [28] | Estevez, G., Smania, D., Yampolsky, M.: Complex bounds for multicritical circle maps with bounded type rotation number. Available at arXiv:2005.02377 |
| [29] | de Faria, E.: Proof of universality for critical circle mappings, Ph.D. Thesis, CUNY (1992) |
| [30] | de Faria, E., On conformal distortion and Sullivan’s sector theorem, Proc. Am. Math. Soc., 126, 67-74 (1998) · Zbl 0899.30016 |
| [31] | de Faria, E., Asymptotic rigidity of scaling ratios for critical circle mappings, Ergod. Theory Dyn. Syst., 19, 995-1035 (1999) · Zbl 0996.37045 |
| [32] | de Faria, E.; Guarino, P., Real bounds and Lyapunov exponents, Disc. Cont. Dyn. Syst. A, 36, 1957-1982 (2016) · Zbl 1364.37091 |
| [33] | de Faria, E., Guarino, P.: Quasisymmetric orbit-flexibility of multicritical circle maps. Available at arXiv:1911.04375 (submitted) · Zbl 1483.37050 |
| [34] | de Faria, E., Guarino, P.: There are no \(\sigma \)-finite absolutely continuous invariant measures for multicritical circle maps. Available at arXiv:2007.10444 (submitted) · Zbl 1483.37050 |
| [35] | de Faria, E.; de Melo, W., Rigidity of critical circle mappings I, J. Eur. Math. Soc., 1, 339-392 (1999) · Zbl 0988.37047 |
| [36] | de Faria, E.; de Melo, W., Rigidity of critical circle mappings II, J. Am. Math. Soc., 13, 343-370 (2000) · Zbl 0988.37048 |
| [37] | de Faria, L.; de Melo, W., Mathematical Tools for One-Dimensional Dynamics (2008), Cambridge: Cambridge University Press, Cambridge · Zbl 1154.30001 |
| [38] | de Faria, E.; de Melo, W.; Pinto, A., Global hyperbolicity of renormalization for \(C^r\) unimodal mappings, Ann. Math., 164, 731-824 (2006) · Zbl 1129.37021 |
| [39] | Feigenbaum, M.; Kadanoff, L.; Shenker, S., Quasi-periodicity in dissipative systems. A renormalization group analysis, Physica, 5D, 370-386 (1982) |
| [40] | Ghys, E.: Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997), ix, xi, 49-95, Panor. Synthèses, 8, Soc. Math. France, Paris (1999) · Zbl 1018.37028 |
| [41] | Gorbovickis, I.; Yampolsky, M., Rigidity, universality, and hyperbolicity of renormalization for critical circle maps with non-integer exponents, Ergod. Theory Dyn. Syst., 40, 1282-1334 (2020) · Zbl 1441.37049 |
| [42] | Graczyk, J.; Sands, D.; Świątek, G., Decay of geometry for unimodal maps: negative Schwarzian case, Ann. Math., 161, 613-677 (2005) · Zbl 1091.37015 |
| [43] | Graczyk, J.; Świątek, G., Singular measures in circle dynamics, Commun. Math. Phys., 157, 213-230 (1993) · Zbl 0792.58025 |
| [44] | Graczyk, J.; Świątek, G., Critical circle maps near bifurcation, Commun. Math. Phys., 176, 227-260 (1996) · Zbl 0857.58034 |
| [45] | Guarino, P.: Rigidity conjecture for \(C^3\) critical circle maps. Ph.D. Thesis. IMPA (2012) |
| [46] | Guarino, P.; Martens, M.; de Melo, W., Rigidity of critical circle maps, Duke Math. J., 167, 2125-2188 (2018) · Zbl 1400.37047 |
| [47] | Guarino, P.; de Melo, W., Rigidity of smooth critical circle maps, J. Eur. Math. Soc., 19, 1729-1783 (2017) · Zbl 1366.37098 |
| [48] | Hall, GR, A \(C^\infty\) Denjoy counterexample, Ergod. Theory Dyn. Syst., 1, 261-272 (1981) · Zbl 0501.58035 |
| [49] | Herman, M., Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49, 5-234 (1979) · Zbl 0448.58019 |
| [50] | Herman, M.: Résultats récents sur la conjugaison différentiable. In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 811-820 (1978) · Zbl 0453.58001 |
| [51] | Herman, M.: 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. www.math.univ-toulouse.fr/ cheritat/Herman/e_,herman.html) |
| [52] | Hu, J.; Sullivan, D., Topological conjugacy of circle diffeomorphisms, Ergod. Theory Dyn. Syst., 17, 173-186 (1997) · Zbl 0997.37010 |
| [53] | Kadanoff, L.; Shenker, S., Critical behavior of a KAM surface. I. Empirical results, J. Stat. Phys., 27, 631-656 (1982) · Zbl 0925.58021 |
| [54] | Katznelson, Y., Sigma-finite invariant measures for smooth mappings of the circle, J. d’Analyse Math., 31, 1-18 (1977) · Zbl 0346.28012 |
| [55] | Katznelson, Y.; Ornstein, D., The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergod. Theory Dyn. Syst., 9, 643-680 (1989) · Zbl 0819.58033 |
| [56] | Katznelson, Y.; Ornstein, D., The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Ergod. Theory Dyn. Syst., 9, 681-690 (1989) · Zbl 0819.58033 |
| [57] | Khanin, K., Universal estimates for critical circle mappings, Chaos, 1, 181-186 (1991) · Zbl 0899.58051 |
| [58] | Khanin, K.: Renormalization and rigidity. In: Proceedings of the International Congress of Mathematicians. Rio de Janeiro, Brazil (2018) · Zbl 1451.37062 |
| [59] | Khanin, K.; Teplinsky, A., Robust rigidity for circle diffeomorphisms with singularities, Invent. Math., 169, 193-218 (2007) · Zbl 1151.37035 |
| [60] | Khanin, K.; Teplinsky, A., Herman’s theory revisited, Invent. Math., 178, 333-344 (2009) · Zbl 1179.37059 |
| [61] | Khinchin, A. Ya.: Continued fractions, (reprint of the 1964 translation), Dover Publications, Inc. (1997) · Zbl 0117.28601 |
| [62] | Khmelev, D.; Yampolsky, M., The rigidity problem for analytic critical circle maps, Mosc. Math. J., 6, 317-351 (2006) · Zbl 1124.37024 |
| [63] | Klein, S., Liu, X.-C., Melo, A.: Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps. Ergodic Theory Dyn. Syst. (to appear) · Zbl 1489.37009 |
| [64] | Lanford, O.E.: Renormalization group methods for critical circle mappings with general rotation number. In: VIIIth International Congress on Mathematical Physics (Marseille, 1986), pp. 532-536. World Scientific. Singapore (1987) |
| [65] | Lanford, O.E.: Renormalization group methods for circle mappings. In: Nonlinear Evolution and Chaotic Phenomena (NATO Adv. Sci. Inst. Ser. B: Phys., 176), pp. 25-36. Plenum, New York (1988) · Zbl 0707.58043 |
| [66] | Lyubich, M., Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Ann. Math., 149, 319-420 (1999) · Zbl 0945.37012 |
| [67] | Lyubich, M.: Teichmüller space of Fibonacci maps. In: Pacifico, M., Guarino, P. (eds.) New Trends in One-Dimensional Dynamics, vol. 285, pp. 221-237. Springer Proceedings in Mathematics & Statistics (2019) · Zbl 1446.37048 |
| [68] | MacKay, RS, A renormalisation approach to invariant circles in area-preserving maps, Physica, 7D, 283-300 (1983) · Zbl 1194.37068 |
| [69] | MacKay, R.S.: Renormalisation in Area-Preserving Maps. In: Advanced Series in Nonlinear Dynamics vol. 6, World-Scientific, Singapore (1993) · Zbl 0791.58002 |
| [70] | Martens, M., The periodic points of renormalization, Ann. Math., 147, 543-584 (1998) · Zbl 0936.37017 |
| [71] | Martens, M.; de Melo, W.; van Strien, S., Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math., 168, 273-318 (1992) · Zbl 0761.58007 |
| [72] | Duff, D. M.c.: \(C^1\)-minimal subsets of the circle. Ann. de l’Institut Fouri. 31, 177-193 (1981) · Zbl 0439.58020 |
| [73] | McMullen, C.T.: Complex dynamics and renormalization, Ann. Math. Stud. 135 (1994) · Zbl 0822.30002 |
| [74] | McMullen, C.T.: Renormalization and 3-manifolds which fiber over the circle, Ann. Math. Stud. 142 (1996) · Zbl 0860.58002 |
| [75] | McMullen, C.T.: Rigidity and inflexibility in conformal dynamics. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 841-855. Doc. Math., Berlin (1998) · Zbl 0914.58015 |
| [76] | de Melo, W.: Rigidity and renormalization in one-dimensional dynamics, In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 765-778. Doc. Math., Berlin (1998) · Zbl 0914.58016 |
| [77] | de Melo, W.; van Strien, S., A structure theorem in one-dimensional dynamics, Ann. Math., 129, 519-546 (1989) · Zbl 0737.58020 |
| [78] | de Melo, W.; van Strien, S., One-Dimensional Dynamics (1993), Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg · Zbl 0791.58003 |
| [79] | Milnor, J.: Dynamics in one complex variable. In: Annals of Mathematics Studies vol. 160, Princeton University Press, Princeton and Oxford (2006) · Zbl 1085.30002 |
| [80] | Misiurewicz, M., Rotation intervals for a class of maps of the real line into itself, Ergod. Theory Dyn. Syst., 6, 117-132 (1986) · Zbl 0615.54030 |
| [81] | Ostlund, S.; Rand, D.; Sethna, J.; Siggia, E., Universal properties of the transition from quasi-periodicity to chaos in dissipative systems, Physica, 8D, 303-342 (1983) · Zbl 0538.58025 |
| [82] | Palmisano, L., A Denjoy counterexample for circle maps with an half-critical point, Math. Z., 280, 749-758 (2015) · Zbl 1359.37094 |
| [83] | Petersen, C.; Zakeri, S., On the Julia set of a typical quadratic polynomial with a Siegel disk, Ann. Math., 159, 2, 1-52 (2004) · Zbl 1069.37038 |
| [84] | Rand, D.: Universality and renormalisation in dynamical systems, In: Bedford, T., Swift, J.W. (eds.) New Directions in Dynamical Systems. Cambridge University Press, Cambridge (1987) · Zbl 0651.58022 |
| [85] | Rand, D.: Global phase space universality, smooth conjugacies and renormalisation: I. The \(C^{1+\alpha }\) case. Nonlinearity 1, 181-202 (1988) · Zbl 0697.58029 |
| [86] | Rand, D., Existence, non-existence and universal breakdown of dissipative golden invariant tori: I. Golden critical circle maps, Nonlinearity, 5, 639-662 (1992) · Zbl 0759.58040 |
| [87] | Shenker, S., Scaling behaviour in a map of a circle onto itself: empirical results, Physica, 5D, 405-411 (1982) |
| [88] | Shishikura, M., The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. Math., 147, 225-267 (1998) · Zbl 0922.58047 |
| [89] | Shishikura, M.: Bifurcation of parabolic fixed points. In: Lei, T. (ed.) The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, pp. 325-364). Cambridge University Press (2000) · Zbl 1062.37043 |
| [90] | Stark, J., Smooth conjugacy and renormalisation for diffeomorphisms of the circle, Nonlinearity, 1, 541-575 (1988) · Zbl 0725.58040 |
| [91] | Sullivan, D., Quasiconformal homeomorphisms and Dynamics I. Fatou-Julia Problem on Wandering Domains, Ann. Math., 122, 401-418 (1985) · Zbl 0589.30022 |
| [92] | Sullivan, D.: Quasiconformal homeomorphisms in dynamics, topology, and geometry. In: Proceedings of the International Congress of Mathematicians, pp. 1216-1228. Berkeley (1986) · Zbl 0698.58030 |
| [93] | Sullivan, D., Bounds, quadratic differentials, and renormalization conjectures, AMS Centen. Pub., II, 417-466 (1992) · Zbl 0936.37016 |
| [94] | Świątek, G., Rational rotation numbers for maps of the circle, Commun. Math. Phys., 119, 109-128 (1988) · Zbl 0656.58017 |
| [95] | Trujillo, F., Hausdorff dimension of invariant measures of multicritical circle maps, Ann. Henri Poincaré, 21, 2861-2875 (2020) · Zbl 1450.37036 |
| [96] | Vieira, A.M.S.: Pares holomorfos e a família de Arnol’d generalizada. Ph.D. Thesis, IME-USP (2015) |
| [97] | Voutaz, E., Hyperbolicity of the renormalization operator for critical \(C^r\) circle mappings, Ergod. Theory Dyn. Syst., 26, 585-618 (2006) · Zbl 1088.37017 |
| [98] | Yampolsky, M., Complex bounds for renormalization of critical circle maps, Ergod. Theory Dyn. Syst., 19, 227-257 (1999) · Zbl 0918.58049 |
| [99] | Yampolsky, M., The attractor of renormalization and rigidity of towers of critical circle maps, Commun. Math. Phys., 218, 537-568 (2001) · Zbl 0978.37033 |
| [100] | Yampolsky, M., Hyperbolicity of renormalization of critical circle maps, Publ. Math. IHES, 96, 1-41 (2002) · Zbl 1030.37027 |
| [101] | Yampolsky, M., Renormalization horseshoe for critical circle maps, Commun. Math. Phys., 240, 75-96 (2003) · Zbl 1078.37034 |
| [102] | Yampolsky, M., Renormalization of unicritical analytic circle maps, C.R. Math. Rep. Acad. Sci. Canada, 39, 77-89 (2017) · Zbl 1380.37085 |
| [103] | Yampolsky, M., Renormalization of bi-cubic circle maps, C. R. Math. Rep. Acad. Sci. Canada, 41, 57-83 (2019) · Zbl 07908820 |
| [104] | Yoccoz, J-C, Il n’y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris, 298, 141-144 (1984) · Zbl 0573.58023 |
| [105] | Yoccoz, J-C, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. Éc. Norm. Sup., 17, 333-359 (1984) · Zbl 0595.57027 |
| [106] | Yoccoz, J.-C.: Continued fraction algorithms for interval exchange maps: an introduction. In: Frontiers in Number Theory, Physics and Geometry, vol 1. On Random Matrices, Zeta Functions and Dynamical Systems. Springer-Verlag (2006) · Zbl 1127.28011 |
| [107] | Zakeri, S., Dynamics of cubic Siegel polynomials, Commun. Math. Phys., 206, 185-233 (1999) · Zbl 0936.37019 |
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