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Lebesgue measure of Feigenbaum Julia sets. (English) Zbl 1494.37026

X. Buff and A. Chéritat [Ann. Math. (2) 176, No. 2, 673–746 (2012; Zbl 1321.37048)] solved a long-standing problem of complex dynamics by constructing quadratic polynomials with Julia sets of positive Lebesgue measure. They constructed three different types of examples: with a Cremer fixed point, with a Siegel disk, or with infinitely many satellite renormalizations.
In this important paper very different examples examples of quadratic polynomials with Julia sets of positive measure are constructed, namely polynomials of Feigenbaum type. In fact, the set of parameters \(c\) for which \(z^2+c\) is a Feigenbaum map with Julia set of positive measure is shown to have Hausdorff dimension at least \(1/2\).

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F25 Renormalization of holomorphic dynamical systems
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

Citations:

Zbl 1321.37048
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References:

[1] Agol, Ian, Tameness of hyperbolic 3-manifolds (2004) · Zbl 1178.57017
[2] Ahlfors, Lars V., Lectures on Quasiconformal Mappings, Univ. Lecture Ser., 38, viii+162 pp. (2006) · Zbl 1103.30001
[3] Avila, Artur; Cheraghi, Davoud, Statistical properties of quadratic polynomials with a neutral fixed point, J. Eur. Math. Soc. (JEMS). Journal of the European Mathematical Society (JEMS), 20, 2005-2062 (2018) · Zbl 1402.37059
[4] Avila, Artur; Lyubich, Mikhail, Hausdorff dimension and conformal measures of {F}eigenbaum {J}ulia sets, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 21, 305-363 (2008) · Zbl 1205.37058
[5] Avila, Artur; Lyubich, Mikhail, The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes, Publ. Math. Inst. Hautes \'{E}tudes Sci.. Publications Math\'{e}matiques. Institut de Hautes \'{E}tudes Scientifiques, 171-223 (2011) · Zbl 1286.37047
[6] Avila, Artur; Moreira, C. G., Hausdorff dimension and the quadratic family
[7] Bara{\'{n}}ski, Krzysztof; Misiurewicz, Micha\l, Omega-limit sets for the {S}tein-{U}lam spiral map, Topology Proc.. Topology Proceedings, 36, 145-172 (2010) · Zbl 1204.37042
[8] Bishop, Christopher J.; Jones, Peter W., Hausdorff dimension and {K}leinian groups, Acta Math.. Acta Mathematica, 179, 1-39 (1997) · Zbl 0921.30032
[9] Bonahon, Francis, Bouts des vari\'{e}t\'{e}s hyperboliques de dimension {\(3\)}, Ann. of Math. (2). Annals of Mathematics. Second Series, 124, 71-158 (1986) · Zbl 0671.57008
[10] Bruin, H.; Keller, G.; Nowicki, T.; van Strein, S., Wild {C}antor attractors exist, Ann. of Math.. Annals of Mathematics. Second Series, 143, 97-130 (1996) · Zbl 0848.58016
[11] Buff, Xavier, Ensembles de {J}ulia de mesure positive (d’apr\`es van {S}trien et {N}owicki). S\'{e}minaire Bourbaki, Vol. 1996/97, Ast\'{e}risque. Ast\'{e}risque, 820-3 (1997) · Zbl 1083.37519
[12] Buff, Xavier; Ch\'{e}ritat, Arnaud, Quadratic {J}ulia sets with positive area, Ann. of Math. (2). Annals of Mathematics. Second Series, 176, 673-746 (2012) · Zbl 1321.37048
[13] Calegari, Danny; Gabai, David, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 19, 385-446 (2006) · Zbl 1090.57010
[14] Canary, Richard D., Ends of hyperbolic \(3\)-manifolds, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 6, 1-35 (1993) · Zbl 0810.57006
[15] De Carvalho, A.; Lyubich, M.; Martens, M., Renormalization in the {H}\'{e}non family. {I}. {U}niversality but non-rigidity, J. Stat. Phys.. Journal of Statistical Physics, 121, 611-669 (2005) · Zbl 1098.37039
[16] Cheraghi, Davoud, Typical orbits of quadratic polynomials with a neutral fixed point: {B}rjuno type, Comm. Math. Phys.. Communications in Mathematical Physics, 322, 999-1035 (2013) · Zbl 1323.37033
[17] Cheraghi, Davoud, Typical orbits of quadratic polynomials with a neutral fixed point: non-{B}rjuno type, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 52, 59-138 (2019) · Zbl 1436.37060
[18] Douady, Adrien, Chirurgie sur les applications holomorphes. Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2, 724-738 (1987)
[19] Douady, Adrien, Disques de {S}iegel et anneaux de {H}erman. S\'{e}minaire Bourbaki, Vol. 1986/87, Ast\'{e}risque, 151-172 (1987) · Zbl 0638.58023
[20] Douady, Adrien, Description of compact sets in {\( \mathbf{C} \)}. Topological {M}ethods in {M}odern {M}athematics. Proceedings of the symposium in honor of {J}ohn {M}ilnor’s sixtieth birthday held at the {S}tate {U}niversity of {N}ew {Y}ork, {S}tony {B}rook, {N}ew {Y}ork, {J}une 14-21, 1991, 429-465 (1993) · Zbl 0801.58025
[21] Douady, Adrien, Does a {J}ulia set depend continuously on the polynomial?. Complex Dynamical Systems, Proc. Sympos. Appl. Math., 49, 91-138 (1994) · Zbl 0934.30023
[22] Douady, Adrien; Buff, Xavier; Devaney, Robert L.; Sentenac, Pierrette, Baby {M}andelbrot sets are born in cauliflowers. The {M}andelbrot {S}et, {T}heme and {V}ariations, London Math. Soc. Lecture Note Ser., 274, 19-36 (2000) · Zbl 1107.37303
[23] Douady, Adrien; Hubbard, John Hamal, \'{E}tude dynamique des polyn\^omes complexes. {P}artie {II}, Publications Math\'{e}matiques d’Orsay, 85, 154 pp. (1985)
[24] Douady, Adrien; Hubbard, John Hamal, On the dynamics of polynomial-like mappings, Ann. Sci. \'{E}cole Norm. Sup. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 18, 287-343 (1985) · Zbl 0587.30028
[25] Dudko, Dzmitry; Lyubich, Mikhail; Selinger, Nikita, Pacman renormalization and self-similarity of the {M}andelbrot set near {S}iegel parameters, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 33, 653-733 (2020) · Zbl 1457.37061
[26] Dudko, Dzmitry; Lyubich, Mikhail, Local connectivity of the {M}andelbrot set at some satellite parameter values of bounded type (2018)
[27] Dudko, Dzmitry; Lyubich, Mikhail, Uniform {\em a priori} bounds for neutral renormalization (2021)
[28] Dudko, Artem; Sutherland, Scott, On the {L}ebesgue measure of the {F}eigenbaum {J}ulia set, Invent. Math.. Inventiones Mathematicae, 221, 167-202 (2020) · Zbl 1454.37045
[29] de Faria, Edson, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 19, 995-1035 (1999) · Zbl 0996.37045
[30] de Faria, Edson; de Melo, Welington, Rigidity of critical circle mappings. {II}, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 13, 343-370 (2000) · Zbl 0988.37048
[31] Eremenko, A. E.; Lyubich, M., Iterations of entire functions, Sov. Math., Dokl.. Soviet Mathematics. Doklady, 30, 592-594 (1984) · Zbl 0588.30027
[32] Eremenko, A. E.; Lyubich, M., Examples of entire functions with pathological dynamics, J. London Math. Soc. (2). Journal of the London Mathematical Society. Second Series, 36, 458-468 (1987) · Zbl 0601.30033
[33] Fatou, P., Sur les \'{e}quations fonctionnelles, Bull. Soc. Math. France. Bulletin de la Soci\'{e}t\'{e} Math\'{e}matique de France, 47, 161-271 (1919) · JFM 47.0921.02
[34] Fatou, P., Notice sur les travaux scientifiques de {M. P. F}atou, astronome adjoint \`a l’{O}bservatoire de {P}aris
[35] Gaidashev, Denis; Yampolsky, Michael, Renormalization of almost commuting pairs, Invent. Math.. Inventiones Mathematicae, 221, 203-236 (2020) · Zbl 1446.37040
[36] Graczyk, Jacek; Smirnov, Stanislav, Non-uniform hyperbolicity in complex dynamics, Invent. Math.. Inventiones Mathematicae, 175, 335-415 (2009) · Zbl 1163.37008
[37] Hofbauer, F.; Keller, G., Quadratic maps without asymptotic measure, Comm. Math. Phys.. Communications in Mathematical Physics, 127, 319-337 (1990) · Zbl 0702.58034
[38] Herman, M., Conjugaison quasi symm\'etrique des diff\'eomorphisms du cercle \`a des rotations et applications aux disques singuliers de {S}iegel (1986)
[39] Hu, J.; Jiang, Y., The {J}ulia set of the {F}eigenbaum quadratic polynomial is locally connected (1993)
[40] Hubbard, John H.; Oberste-Vorth, Ralph W., H\'{e}non mappings in the complex domain. {I}. {T}he global topology of dynamical space, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 5-46 (1994) · Zbl 0839.54029
[41] Inou, H.; Shishikura, M., The renormalization for parabolic fixed points and their perturbation (2008)
[42] Jiang, Yunping, Infinitely renormalizable quadratic polynomials, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 352, 5077-5091 (2000) · Zbl 0947.37029
[43] Kahn, Jeremy, A priori bounds for some infinitely renormalizable quadratics: {I. B}ounded primitive combinatorics (2006)
[44] Kahn, Jeremy; Lyubich, Mikhail, A priori bounds for some infinitely renormalizable quadratics. {II}. {D}ecorations, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 41, 57-84 (2008) · Zbl 1156.37311
[45] Lavaurs, P., Syst\`emes dynamiques holomorphes: explosion des points p\'eriodiques paraboliques. (1989)
[46] Lyubich, M. Yu., Measurable dynamics of the exponential, Syberian J. Math.. Akademiya Nauk SSSR. Sibirskoe Otdelenie. Sibirski\u{\i} Matematicheski\u{\i} Zhurnal, 28, 111-127 (1987) · Zbl 0667.58037
[47] Lyubich, M. Yu., Typical behavior of trajectories of the rational mapping of a sphere, Dokl. Akad. Nauk SSSR. Doklady Akademii Nauk SSSR. Caroline Series, 268, 29-32 (1983) · Zbl 0595.30034
[48] Lyubich, Mikhail, How big is the set of infinitely renormalizable quadratics?. Voronezh {W}inter {M}athematical {S}chools, Amer. Math. Soc. Transl. Ser. 2, 184, 131-143 (1998) · Zbl 0910.58033
[49] Lyubich, Mikhail, Feigenbaum-{C}oullet-{T}resser universality and {M}ilnor’s hairiness conjecture, Ann. of Math. (2). Annals of Mathematics. Second Series, 149, 319-420 (1999) · Zbl 0945.37012
[50] book in preparation; available on author’s webpage, Conformal {G}eometry and {D}ynamics of {Q}uadratic {P}olynomials (2030)
[51] Lyubich, Mikhail, On the {L}ebesgue measure of the {J}ulia setof a quadratic polynomial
[52] Lyubich, Mikhail, On the borderline of real and complex dynamics. Proceeedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2 ({Z}\"{u}rich, 1994), 1203-1215 (1995) · Zbl 0847.58021
[53] Lyubich, Mikhail; Minsky, Y., Laminations in holomorphic dynamics, J. Differential Geom.. Journal of Differential Geometry, 47, 17-94 (1997) · Zbl 0910.58032
[54] Lyubich, Mikhail; Yampolsky, M., Dynamics of quadratic polynomials: complex bounds for real maps, Ann. Inst. Fourier (Grenoble). Univerist\'{e} de Grenoble. Annales de l’Institut Fourier, 47, 1219-1255 (1997) · Zbl 0881.58053
[55] Mattila, Pertti, Geometry of Sets and Measures in {E}uclidean Spaces, Cambridge Stud. Adv. Math., 44, xii+343 pp. (1995) · Zbl 0819.28004
[56] McMullen, Curt, Area and {H}ausdorff dimension of {J}ulia sets of entire functions, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 300, 329-342 (1987) · Zbl 0618.30027
[57] McMullen, Curtis T., Complex {D}ynamics and {R}enormalization, Annals of Mathematics Studies, 135, x+214 pp. (1994) · Zbl 0822.30002
[58] McMullen, Curtis T., Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, 142, x+253 pp. (1996) · Zbl 0860.58002
[59] McMullen, Curtis T., Self-similarity of {S}iegel disks and {H}ausdorff dimension of {J}ulia sets, Acta Math.. Acta Mathematica, 180, 247-292 (1998) · Zbl 0930.37022
[60] de Melo, Welington; van Strien, Sebastian, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb., 25, xiv+605 pp. (1993) · Zbl 0791.58003
[61] Petersen, Carsten Lunde, Local connectivity of some {J}ulia sets containing a circle with an irrational rotation, Acta Math.. Acta Mathematica, 177, 163-224 (1996) · Zbl 0884.30020
[62] Prado, E. A., Ergodicity of conformal measures for unimodal polynomials, Conform. Geom. Dyn.. 2, 29-44 · Zbl 0893.58046
[63] Prado, Eduardo A., Ergodicity of conformal measures for unimodal polynomials, Conform. Geom. Dyn.. Conformal Geometry and Dynamics. An Electronic Journal of the American Mathematical Society, 2, 29-44 (1998) · Zbl 0893.58046
[64] Przytycki, Feliks; Rohde, Steffen, Porosity of {C}ollet-{E}ckmann {J}ulia sets, Fund. Math.. Fundamenta Mathematicae, 155, 189-199 (1998) · Zbl 0908.58054
[65] Rees, Mary, The exponential map is not recurrent, Math. Z.. Mathematische Zeitschrift, 191, 593-598 (1986) · Zbl 0595.30033
[66] Shishikura, Mitsuhiro, The {H}ausdorff dimension of the boundary of the {M}andelbrot set and {J}ulia sets, Ann. of Math. (2). Annals of Mathematics. Second Series, 147, 225-267 (1998) · Zbl 0922.58047
[67] Shishikura, Mitsuhiro, Topological, geometric and complex analytic properties of {J}ulia sets. Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2, 886-895 (1995) · Zbl 0843.30026
[68] Sullivan, Dennis, Growth of positive harmonic functions and {K}leinian group limit sets of zero planar measure and {H}ausdorff dimension two. Geometry {S}ymposium, {U}trecht 1980, Lecture Notes in Math., 894, 127-144 (1981) · Zbl 0465.00016
[69] Sullivan, Dennis, Conformal dynamical systems. Geometric {D}ynamics, Lecture Notes in Math., 1007, 725-752 (1983) · Zbl 0524.58024
[70] Sullivan, Dennis, Bounds, quadratic differentials, and renormalization conjectures. American {M}athematical {S}ociety {C}entennial {P}ublications, {V}ol. {II}, 417-466 (1992) · Zbl 0936.37016
[71] {\'{S}}wiatek, Grzegorz, On critical circle homeomorphisms, Bol. Soc. Brasil. Mat. (N.S.). Boletim da Sociedade Brasileira de Matem\'{a}tica. Nova S\'{e}rie, 29, 329-351 (1998) · Zbl 1053.37019
[72] Thurston, W., The geometry and topology of 3-manifolds (1982)
[73] Urba\'{n}ski, Mariusz, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 14, 391-414 (1994) · Zbl 0807.58025
[74] Urba\'{n}ski, Mariusz; Zdunik, Anna, Geometry and ergodic theory of non-hyperbolic exponential maps, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 359, 3973-3997 (2007) · Zbl 1110.37038
[75] Yampolsky, Michael, Complex bounds for renormalization of critical circle maps, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 19, 227-257 (1999) · Zbl 0918.58049
[76] Yampolsky, Michael, Hyperbolicity of renormalization of critical circle maps, Publ. Math. Inst. Hautes \'{E}tudes Sci.. Publications Math\'{e}matiques. Institut de Hautes \'{E}tudes Scientifiques, 1-41 (2002) · Zbl 1030.37027
[77] Yampolsky, Michael, Siegel disks and renormalization fixed points. Holomorphic dynamics and renormalization, Fields Inst. Commun., 53, 377-393 (2008) · Zbl 1157.37321
[78] Yarrington, Brian William, Local Connectivity and {L}ebesgue Measure of Polynomial {J}ulia Sets, 94 pp. (1995)
[79] Yoccoz, Jean-Christophe, Il n’y a pas de contre-exemple de {D}enjoy analytique, C. R. Acad. Sci. Paris S\'{e}r. I Math.. Comptes Rendus des S\'{e}ances de l’Acad\'{e}mie des Sciences. S\'{e}rie I. Math\'{e}matique, 298, 141-144 (1984) · Zbl 0573.58023
[80] Zakharevich, M. I., The behavior of trajectories and the ergodic hypothesis for quadratic mappings of a simplex, Uspekhi Mat. Nauk. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 33, 207-208 (1978) · Zbl 0407.58030
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