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Discontinuity, Nonlinearity, and Complexity 15(4) (2026) 525--539 | DOI:10.5890/DNC.2026.12.004

Gamaliel Blé$^1$, Carlos Antonio Marin-Mendoza$^2$, Rogelio Valdez Delgado$^{2}$

$^1$ División Académica de Ciencias Básicas. Universidad Juárez Autónoma de Tabasco, Km 1, Carretera Cunduacán–Jalpa de Méndez, Cunduacán, 86690, Tabasco, México

$^2$ Centro de Investigación en Ciencias, Instituto de Investigación en Ciencias Básicas y Aplicadas. Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, Cuernavaca, 62209, Morelos, México

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Abstract

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 $\theta_*$.

Acknowledgments

Carlos Antonio Marin-Mendoza (CVU-714167) thanks to the Secretaría de Ciencia, Humanidades, Tecnología e Innovación (SECIHTI) for its support during the writing of this paper.

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