×

Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters. (English) Zbl 1457.37061

In this deep and extremely technical paper, the authors are concerned with the renormalization of certain analytic maps. They design a new renormalization (too involved to be described here in a few lines), and establish two main results.
The first one states the hyperbolicity of the corresponding renormalization operator. The authors remark that the surgery applied by B. Branner and A. Douady [Lect. Notes Math. 1345, 11–72 (1988; Zbl 0668.58026)] is the prototype for the pacman renormalization developed in the paper.
The second main result is a scaling theorem which confirms the prediction of the drawn pictures on the self-similarity features near to main cardioid of the Mandelbrot set \(\mathcal{M}\), so their renormalization theory explains the phenomenon. To be more precise, let \(\Theta_{\mathrm{per}}\) be the set of rotation numbers periodic under the map \(\theta\mapsto \theta/(1-\theta)\) if \(0\leq \theta \leq 1/2,\) and \(\theta\mapsto (2\theta - 1)/ \theta\) if \(1/2\leq \theta \leq 1;\) then, if \(\theta\in\Theta_{\mathrm{per}}\) and \(p_n/q_n\) are its continued fraction approximands, then \(\frac{|c(\theta)-a_{p_n/q_n}|}{q^2_n}\) tends to a non-zero constant when \(n\) goes to infinity, where \(c(\theta)\) denotes the parametrization of the main cardioid by the rotation number, and \(a_{p/q}\) denotes the center of a satellite hyperbolic component of \(\mathcal{M}\) attached to the parabolic point \(c(p/q)\).
Despite its difficulty, the paper is well structured and the reader is always suitably guided in the comprehension of the different parts. It is worth mentioning that the paper includes some open questions which may attract the attention for further developments in this direction, as the self-similarity of limbs of the Mandelbrot set \(\mathcal{M}\) or the local connectivity of \(\mathcal{M}\) at some satellite parameters of bounded type.

MSC:

37F25 Renormalization of holomorphic dynamical systems
37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems
37F46 Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets
37F44 Holomorphic families of dynamical systems; holomorphic motions; semigroups of holomorphic maps
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37E45 Rotation numbers and vectors

Citations:

Zbl 0668.58026
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 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., 114, 171-223 (2011) · Zbl 1286.37047 · doi:10.1007/s10240-011-0034-2
[2] A. Avila and M. Lyubich, Lebesgue measure of Feigenbaum Julia sets, arXiv:1504.02986. · Zbl 1173.37045
[3] Buff, Xavier; Ch\'{e}ritat, Arnaud, Quadratic Julia sets with positive area, Ann. of Math. (2), 176, 2, 673-746 (2012) · Zbl 1321.37048 · doi:10.4007/annals.2012.176.2.1
[4] Branner, Bodil; Douady, Adrien, Surgery on complex polynomials. Holomorphic dynamics, Mexico, 1986, Lecture Notes in Math. 1345, 11-72 (1988), Springer, Berlin · Zbl 0668.58026 · doi:10.1007/BFb0081395
[5] Bers, Lipman; Royden, H. L., Holomorphic families of injections, Acta Math., 157, 3-4, 259-286 (1986) · Zbl 0619.30027 · doi:10.1007/BF02392595
[6] A. Cheritat, Near parabolic renormalization for unicritical holomorphic maps, arXiv:1404.4735.
[7] Childers, Douglas K., Are there critical points on the boundaries of mother hedgehogs?. Holomorphic dynamics and renormalization, Fields Inst. Commun. 53, 75-87 (2008), Amer. Math. Soc., Providence, RI · Zbl 1153.37024
[8] Douady, Adrien, Disques de Siegel et anneaux de Herman, Ast\'{e}risque, 152-153, 4, 151-172 (1988) (1987) · Zbl 0638.58023
[9] Douady, Adrien, Does a Julia set depend continuously on the polynomial?. Complex dynamical systems, Cincinnati, OH, 1994, Proc. Sympos. Appl. Math. 49, 91-138 (1994), Amer. Math. Soc., Providence, RI · Zbl 0934.30023 · doi:10.1090/psapm/049/1315535
[10] A. Douady and J. H. Hubbard, \'Etude dynamique des polyn\^omes complexes, Publication Mathematiques d’Orsay, 84-02 and 85-04.
[11] Douady, Adrien; Hubbard, John Hamal, On the dynamics of polynomial-like mappings, Ann. Sci. \'{E}cole Norm. Sup. (4), 18, 2, 287-343 (1985) · Zbl 0587.30028
[12] D. Dudko and M. Lyubich, Local connectivity of the Mandelbrot set at some satellite parameters of bounded type, arXiv:1808.10425.
[13] Graczyk, Jacek; Jones, Peter, Dimension of the boundary of quasiconformal Siegel disks, Invent. Math., 148, 3, 465-493 (2002) · Zbl 1079.37507 · doi:10.1007/s002220100198
[14] D. Gaidashev and M. Yampolsky, Renormalization of almost commuting pairs, Invent.math. (2020). https://doi.org/10.1007/s00222-020-00947-w. · Zbl 1446.37040
[15] M. Herman, Conjugaison quasi symm\'etrique des diff\'eomorphisms du cercle \`a des rotations et applications aux disques singuliers de Siegel, Manuscript, 1986.
[16] Hirsch, M. W.; Pugh, C. C.; Shub, M., Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, ii+149 pp. (1977), Springer-Verlag, Berlin-New York · Zbl 0355.58009
[17] H. Inou and M. Shishikura, The renormalization for parabolic fixed points and their perturbations, Manuscript, 2008.
[18] P. Lavaurs, Syst\`emes dynamiques holomorphes: explosion des points p\'eriodiques paraboliques, The\`se, Univirsit\'e Paris-Sud, 1989.
[19] Lyubich, Mikhail, Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Ann. of Math. (2), 149, 2, 319-420 (1999) · Zbl 0945.37012 · doi:10.2307/120968
[20] M. Lyubich, Conformal geometry and dynamics of quadratic polynomials, in preparation, www.math.stonybrook.edu/ mlyubich/book.pdf.
[21] McMullen, Curtis T., Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies 142, x+253 pp. (1996), Princeton University Press, Princeton, NJ · Zbl 0860.58002 · doi:10.1515/9781400865178
[22] McMullen, Curtis T., Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math., 180, 2, 247-292 (1998) · Zbl 0930.37022 · doi:10.1007/BF02392901
[23] Manton, N. S.; Nauenberg, M., Universal scaling behaviour for iterated maps in the complex plane, Comm. Math. Phys., 89, 4, 555-570 (1983) · Zbl 0523.30018
[24] MacKay, R. S.; Percival, I. C., Universal small-scale structure near the boundary of Siegel disks of arbitrary rotation number, Phys. D, 26, 1-3, 193-202 (1987) · Zbl 0612.58028 · doi:10.1016/0167-2789(87)90223-5
[25] Petersen, Carsten Lunde, Local connectivity of some Julia sets containing a circle with an irrational rotation, Acta Math., 177, 2, 163-224 (1996) · Zbl 0884.30020 · doi:10.1007/BF02392621
[26] P\'{e}rez-Marco, Ricardo, Fixed points and circle maps, Acta Math., 179, 2, 243-294 (1997) · Zbl 0914.58027 · doi:10.1007/BF02392745
[27] J. Riedl, Arcs in multibrot sets, locally connected Julia sets and their construction by quasiconformal surgery, Ph.D. thesis, TU M\"unchen, 2000. IMS Thesis Server, http://www.math.stonybrook.edu/ims-thesis-server · Zbl 1121.37314
[28] Sullivan, Dennis, Bounds, quadratic differentials, and renormalization conjectures. American Mathematical Society centennial publications, Vol. II, Providence, RI, 1988, 417-466 (1992), Amer. Math. Soc., Providence, RI · Zbl 0936.37016
[29] Stirnemann, Andreas, Existence of the Siegel disc renormalization fixed point, Nonlinearity, 7, 3, 959-974 (1994) · Zbl 0803.47056
[30] Sullivan, Dennis P.; Thurston, William P., Extending holomorphic motions, Acta Math., 157, 3-4, 243-257 (1986) · Zbl 0619.30026 · doi:10.1007/BF02392594
[31] \'{S}wi\polhk atek, Grzegorz, On critical circle homeomorphisms, Bol. Soc. Brasil. Mat. (N.S.), 29, 2, 329-351 (1998) · Zbl 1053.37019 · doi:10.1007/BF01237654
[32] Widom, Michael, Renormalization group analysis of quasiperiodicity in analytic maps, Comm. Math. Phys., 92, 1, 121-136 (1983) · Zbl 0535.58037
[33] Yampolsky, Michael, Siegel disks and renormalization fixed points. Holomorphic dynamics and renormalization, Fields Inst. Commun. 53, 377-393 (2008), Amer. Math. Soc., Providence, RI · Zbl 1157.37321
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.