On perturbations of rational maps and construction of semiconjugacies on the Julia sets (Studies on complex dynamics and related topics)
Year: 2001
Type: article
Abstract: In this note, we investigate perturbations of parabolic rational maps on the Riemann sphere, and dynamical stability of their Julia sets.A rational map $f$ is called parabolic if every critical point is contained in the Fatou set.If aperturbation of $f$ into another parabolic rational map is horocyclic, then we can construct asemiconjugacy on their Julia sets.This means that parabolic rational maps have weak J-stability 1J-stability Let $f$ : $\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$ be arational map of degree $d\geq 2$ , and $\mathrm{R}\mathrm{a}\mathrm{t}_{d}$ the space of all rational maps of degree $d$ .The topology of this space is defined by the uniform convergence on the sphere measured by the spherical distance $d_{\sigma}(\cdot, \cdot)$ .In this note, we discuss perturbations of a rational map (especially parabolic rational map) $f$ within $\mathrm{R}\mathrm{a}\mathrm{t}_{\mathrm{d}}$ , and study the dynamical stability of $f$ on the Julia set $J(f)$ :That is, structural stability of $f$ restricted on the Julia set.Here perturbation of $f$ means a family of rational maps $\{f_{\epsilon}\in \mathrm{R}\mathrm{a}\mathrm{t}_{d} : \epsilon\in[0,1]\}$ satisfying $f_{0}=f$ and $d_{\sigma}(f_{\epsilon}, f)arrow \mathrm{O}(\epsilonarrow 0)$ .We represent this family as the form of convergence, $f_{\epsilon}arrow f$ .Aboutthis, the result below is famous: Theorem 1.1 ( $\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}$ -Sad-Sullivan[10])1 If f has a connected neighbor- hood U $\subset \mathrm{R}\mathrm{a}\mathrm{t}_{d}$ w here the number of attracting cycles is locally constant, then
Author: Tomoki Kawahira
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W629100739