Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters

Abstract

In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of "Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. Then we prove that these periodic points are hyperbolic with one-dimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters. Contents 1. Introduction 1 2. Pacman renormalization operator 8 3. Siegel pacmen 15 4. Control of pullbacks 26 5. Maximal prepacmen 41 6. Maximal parabolic prepacmen 45 7. Hyperbolicity Theorem 55 8. Scaling Theorem 57 Appendix A. Sector renormalizations of a rotation 60 Appendix B. Lifting of curves under anti-renormalization 63 Appendix C. The molecule renormalization 81 References 85

Local connectivity of the Mandelbrot set at some satellite parameters of bounded type

“…Consider a rational number r = p/q close to θ and let f r be the parabolic pacman with rotation number r. An attracting basin of α(f r ) admits the globalization H in the dynamical plane of F r , see §3.2.14. It was shown in [DLS,§6] that H contains the critical orbit. Periodic components of H are attached to α(F r ) and are enumerated as (H i ) i∈Z from left-to-right with H 0 0.…”

“…realizing the first return of points in X − ∪ X + back to X; and • the gluing map ψ : projecting (2.4) to a new rotation L µ , where ω is the angle of X at 0. A sector renormalization is an iteration of the prime renormalization (see [DLS,Lemma A.2]); in particular, µ = R m prm (θ) for some m ≥ 1.…”

“…Similarly, the p/q-antirenormalization is defined for any rational number p/q, see [DLS,§ B.1.3]. The 1/3 and 2/3 antirenormalization are called prime.…”

“…The 1/3 and 2/3 antirenormalization are called prime. Any other antirenormalization is an iteration of prime antirenormalizations, see [DLS,Lemma B.3].…”

“…Equivalently, the canonical homeomorphism h : D → W can be characterized using unique extension along curves as in [DLS,Theorem B.8]. A similar characterization has a canonical homeomorphism between ∆ 0 and ∆ new 0 .…”