Typical orbits of quadratic polynomials with a neutral fixed point:  Non-Brjuno type

Cheraghi, Davoud

doi:10.24033/asens.2384

Typical orbits of quadratic polynomials with a neutral fixed point: Non-Brjuno type
[Orbites typiques des polynômes quadratiques avec un point fixe neutre: type non-Brjuno]

Cheraghi, Davoud

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 59-138.

Résumé
Abstract

On étudie les aspects quantitatifs et analytiques du procédé de renormalisation presque parabolique introduit par Inou et Shishikura en 2006. Ceci fournit des techniques pour étudier la dynamique de certaines applications holomorphes de la forme $f (z) = e^{2 π i α} z + 𝒪 (z^{2})$ , dont les polynômes quadratiques $e^{2 π i α} z + z^{2}$ , pour certaines valeurs irrationnelles de $α$ . Les principaux résultats de cet article concernent les propriétés à petite échelle des attracteurs au sens de la théorie de la mesure pour ces applications ainsi que de leur dépendance en fonction des données du problème. On obtient également une borne supérieure optimale sur la taille du domaine maximal de linéarisation en termes de la série de Brjuno-Siegel-Yoccoz de $α$ .

We investigate the quantitative and analytic aspects of the near-parabolic renormalization scheme introduced by Inou and Shishikura in 2006. These provide techniques to study the dynamics of some holomorphic maps of the form $f (z) = e^{2 π i α} z + 𝒪 (z^{2})$ , including the quadratic polynomials $e^{2 π i α} z + z^{2}$ , for some irrational values of $α$ . The main results of the paper concern fine-scale features of the measure-theoretic attractors of these maps, and their dependence on the data. As a bi-product, we establish an optimal upper bound on the size of the maximal linearization domain in terms of the Siegel-Brjuno-Yoccoz series of $α$ .

DOI : 10.24033/asens.2384

Classification : 37F50; 46T25, 37F25
Keywords: Small divisors, Cremer fixed points, post-critical set, near-parabolic renormalization
Mot clés : Petits diviseurs, points fixes de Cremer, ensemble post-critique, renormalisation presque parabolique

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     author = {Cheraghi, Davoud},
     title = {Typical orbits of quadratic polynomials with a neutral fixed point:  {Non-Brjuno} type},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {59--138},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {1},
     year = {2019},
     doi = {10.24033/asens.2384},
     mrnumber = {3940907},
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     url = {https://www.numdam.org/articles/10.24033/asens.2384/}
}

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Cheraghi, Davoud. Typical orbits of quadratic polynomials with a neutral fixed point:  Non-Brjuno type. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 59-138. doi : 10.24033/asens.2384. https://www.numdam.org/articles/10.24033/asens.2384/

Bibliographie
Cité par

Avila, A.; Buff, X.; Chéritat, A. Siegel disks with smooth boundaries, Acta Math., Volume 193 (2004), pp. 1-30 (ISSN: 0001-5962) | DOI | MR | Zbl

Avila, A.; Cheraghi, D. Statistical properties of quadratic polynomials with a neutral fixed point, J. Eur. Math. Soc., Volume 20 (2018), pp. 2005-2062 (ISSN: 1435-9855) | DOI | MR

Avila, A.; Lyubich, M. Y. Lebesgue measure of Feigenbaum Julia sets (preprint arXiv:1504.02986 ) | MR

Blokh, A.; Buff, X.; Chéritat, A.; Oversteegen, L. The solar Julia sets of basic quadratic Cremer polynomials, Ergodic Theory Dynam. Systems, Volume 30 (2010), pp. 51-65 (ISSN: 0143-3857) | DOI | MR | Zbl

Buff, X.; Chéritat, A. Upper bound for the size of quadratic Siegel disks, Invent. math., Volume 156 (2004), pp. 1-24 (ISSN: 0020-9910) | DOI | MR | Zbl

Buff, X.; Chéritat, A. How regular can the boundary of a quadratic Siegel disk be?, Proc. Amer. Math. Soc., Volume 135 (2007), pp. 1073-1080 (ISSN: 0002-9939) | DOI | MR | Zbl

Buff, X.; Chéritat, A. Quadratic Julia sets with positive area, Ann. of Math., Volume 176 (2012), pp. 673-746 (ISSN: 0003-486X) | DOI | MR | Zbl

Brjuno, A. D. Analytic form of differential equations. I, II, Trudy Moskov. Mat. Obšč., Volume 25 (1971) (ISSN: 0134-8663) | MR | Zbl

Cheraghi, D.; Chéritat, A. A proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type, Invent. math., Volume 202 (2015), pp. 677-742 (ISSN: 0020-9910) | DOI | MR

Chierchia, L.; Falcolini, C. Compensations in small divisor problems, Comm. Math. Phys., Volume 175 (1996), pp. 135-160 http://projecteuclid.org/euclid.cmp/1104275733 (ISSN: 0010-3616) | DOI | MR | Zbl

Chéritat, A. Relatively compact Siegel disks with non-locally connected boundaries, Math. Ann., Volume 349 (2011), pp. 529-542 (ISSN: 0025-5831) | DOI | MR | Zbl

Cheraghi, D. Typical orbits of quadratic polynomials with a neutral fixed point: Brjuno type, Comm. Math. Phys., Volume 322 (2013), pp. 999-1035 (ISSN: 0010-3616) | DOI | MR

Childers, D. K., Holomorphic dynamics and renormalization (Fields Inst. Commun.), Volume 53, Amer. Math. Soc., 2008, pp. 75-87 | MR | Zbl

Cremer, H. Über die Häufigkeit der Nichtzentren, Math. Ann., Volume 115 (1938), pp. 573-580 (ISSN: 0025-5831) | DOI | JFM | MR | Zbl

Cheraghi, D.; Shishikura, M. Satellite renormalization of quadratic polynomials (preprint arXiv:1509.07843 )

Douady, A. Disques de Siegel et anneaux de Herman, Séminaire Bourbaki, vol. 1986/1987, exposé n^o 677, Astérisque, Volume 152-153 (1987), pp. 151-172 (ISSN: 0303-1179) | MR | Zbl

Douady, A., Complex dynamical systems (Cincinnati, OH, 1994) (Proc. Sympos. Appl. Math.), Volume 49, Amer. Math. Soc., 1994, pp. 91-138 | DOI | MR | Zbl

Duren, P. L., Grundl. math. Wiss., 259, Springer, 1983, 382 pages (ISBN: 0-387-90795-5) | MR | Zbl

Geyer, L. Siegel discs, Herman rings and the Arnold family, Trans. Amer. Math. Soc., Volume 353 (2001), pp. 3661-3683 (ISSN: 0002-9947) | DOI | MR | Zbl

Graczyk, J.; Jones, P. Dimension of the boundary of quasiconformal Siegel disks, Invent. math., Volume 148 (2002), pp. 465-493 (ISSN: 0020-9910) | DOI | MR | Zbl

Graczyk, J.; Świątek, G. Siegel disks with critical points in their boundaries, Duke Math. J., Volume 119 (2003), pp. 189-196 (ISSN: 0012-7094) | DOI | MR | Zbl

Herman, M. R. Are there critical points on the boundaries of singular domains?, Comm. Math. Phys., Volume 99 (1985), pp. 593-612 | DOI | MR | Zbl

Herman, M. R. Recent results and some open questions on Siegel's linearization theorem of germs of complex analytic diffeomorphisms of $𝐂^{n}$ near a fixed point, VIIIth International congress on mathematical physics (Marseille, 1986), World Sci. Publishing (1987), pp. 138-184 | MR

Inou, H.; Shishikura, M. The renormalization for parabolic fixed points and their perturbation (2006) (preprint https://www.math.kyoto-u.ac.jp/~mitsu/pararenorm/ )

Lyubich, M. Y. Some typical properties of the dynamics of rational mappings, Uspekhi Mat. Nauk, Volume 38 (1983), pp. 197-198 (ISSN: 0042-1316) | MR | Zbl

Lyubich, M. Y. Typical behavior of trajectories of the rational mapping of a sphere, Dokl. Akad. Nauk SSSR, Volume 268 (1983), pp. 29-32 (ISSN: 0002-3264) | MR | Zbl

McMullen, C. T. Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math., Volume 180 (1998), pp. 247-292 (ISSN: 0001-5962) | DOI | MR | Zbl

Mañé, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps, Ann. Sci. École Norm. Sup., Volume 16 (1983), pp. 193-217 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Okuyama, Y. Nevanlinna, Siegel, and Cremer, Indiana Univ. Math. J., Volume 53 (2004), pp. 755-763 (ISSN: 0022-2518) | DOI | MR | Zbl

Pérez, R. A. A brief but historic article of Siegel, Notices Amer. Math. Soc., Volume 58 (2011), pp. 558-566 (ISSN: 0002-9920) | MR | Zbl

Pérez Marco, R. Sur les dynamiques holomorphes non linéarisables et une conjecture de V. I. Arnolʼd, Ann. Sci. École Norm. Sup., Volume 26 (1993), pp. 565-644 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Pérez-Marco, R. Fixed points and circle maps, Acta Math., Volume 179 (1997), pp. 243-294 (ISSN: 0001-5962) | DOI | MR | Zbl

Pfeiffer, G. A. On the conformal mapping of curvilinear angles. The functional equation $ϕ [f (x)] = a_{1} ϕ (x)$ , Trans. Amer. Math. Soc., Volume 18 (1917), pp. 185-198 (ISSN: 0002-9947) | DOI | JFM | MR

Pommerenke, C., Vandenhoeck & Ruprecht, Göttingen, 1975, 376 pages | MR | Zbl

Petersen, C. L.; Zakeri, S. On the Julia set of a typical quadratic polynomial with a Siegel disk, Ann. of Math., Volume 159 (2004), pp. 1-52 (ISSN: 0003-486X) | DOI | MR | Zbl

Shishikura, M., The Mandelbrot set, theme and variations (London Math. Soc. Lecture Note Ser.), Volume 274, Cambridge Univ. Press, 2000, pp. 325-363 | DOI | MR | Zbl

Shishikura, M. The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math., Volume 147 (1998), pp. 225-267 (ISSN: 0003-486X) | DOI | MR | Zbl

Siegel, C. L. Iteration of analytic functions, Ann. of Math., Volume 43 (1942), pp. 607-612 (ISSN: 0003-486X) | DOI | MR | Zbl

Sullivan, D. Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) (Lecture Notes in Math.), Volume 1007, Springer (1983), pp. 725-752 | DOI | MR | Zbl

Yampolsky, M., Holomorphic dynamics and renormalization (Fields Inst. Commun.), Volume 53, Amer. Math. Soc., 2008, pp. 377-393 | MR | Zbl

Yoccoz, J.-C. Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque, Volume 231 (1995), pp. 3-88 (ISSN: 0303-1179) | MR

Zakeri, S. On Siegel disks of a class of entire maps, Duke Math. J., Volume 152 (2010), pp. 481-532 (ISSN: 0012-7094) | DOI | MR | Zbl

Zhang, G. All bounded type Siegel disks of rational maps are quasi-disks, Invent. math., Volume 185 (2011), pp. 421-466 (ISSN: 0020-9910) | DOI | MR | Zbl

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