Pacman Renormalization on Siegel Parameters with Rotation Number of Periodic Type in the Quadratic Family
DOI:
https://doi.org/10.5890/DNC.2026.12.004Abstract
In this paper, we proved that there exists a universal constant of convergence rate when a Siegel map whose rotation number $\theta_*$ has a periodic continued fraction, on the boundary of any hyperbolic component of any quadratic-like family, is approximated by functions with a parabolic or an attracting cycle in the family of quadratic-like maps. Additionally, the satellite's valuable flowers of the functions converging to the Siegel map also approximate the Siegel disk. In particular, there is a universal convergence rate in the Mandelbrot set when Siegel parameters of any period and rotation number $\theta_*$ are approximated by the centers of hyperbolic components, which are related to some of the denominators of the continued fraction that converge to $\theta_*$.References
[1] McMullen, C.T. (1998), Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Mathematica, 180(2), 247-292.
[2] Branner, B. and Douady, A. (1988), Surgery on complex polynomials, In: Gomez-Mont, X., Seade, J.A., and Verjovski, A. (eds) Holomorphic Dynamics, Lecture Notes in Mathematics, 1345, Springer, Berlin, Heidelberg.
[3] Yampolsky, M. (2008), Siegel disks and renormalization fixed points, Holomorphic Dynamics and Renormalization, Fields Institute Communications, 53, American Mathematical Society, Providence, RI, 377-393.
[4] Dudko, D., Lyubich, M., and Selinger, N. (2020), Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters, Journal of the American Mathematical Society, 33(3), 653-733.
[5] Marin-Mendoza, C. and Valdez, R. (2023), Pacman renormalization in Siegel parameters of bounded type, Advances in Pure Mathematics, 13, 674-693.
[6] Dudko, D. and Lyubich, M. (2023), Local connectivity of the Mandelbrot set at some satellite parameters of bounded type, Geometric and Functional Analysis, 33, 912-1047.
[7] Lyubich, M. (1999), Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture, Annals of Mathematics, 149(2), 319-420.
[8] Douady, A. and Hubbard, J. (1985), On the dynamics of polynomial-like mappings, Annales scientifiques de l'École normale supérieure, 18.2, 287-343.
[9] Widž, J. (2012), From the history of continued fractions, WDS'09 Proceedings of Contributed Papers, Part I, 176-181.
[10] Douady, A. (1994), Does a Julia set depend continuously on the polynomial? Complex Dynamical Systems (Cincinnati, OH, 1994), Proceedings of Symposia in Applied Mathematics, 49, American Mathematical Society, Providence, RI, 1994, 91-138.
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