Pacman Renormalization on Siegel Parameters with Rotation Number of Periodic Type in the Quadratic Family - Blé, Gamaliel; Marin-Mendoza, Carlos Antonio; Delgado, Rogelio Valdez (Discontinuity, Nonlinearity, and Complexity, 2026)
In this paper, we proved that there exists a universal constant of convergence rate when a Siegel map whose rotation number θ_* 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 θ_* are approximated by the centers of hyperbolic components, which are related to some of the denominators of the continued fraction that converge to θ_*.