A priori bounds for some infinitely renormalizable quadratics: II. Decorations

A Priori Bounds for Some Infinitely Renormalizable Quadratics: Ii. Decorations

A Priori Bounds for Some Infinitely Renormalizable Quadratics: Ii. Decorations

A. – A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of deco...

A Priori Bounds for Some Infinitely Renormalizable Quadratics: I. Bounded Primitive Combinatorics

1 2 Fe b 20 07 A PRIORI BOUNDS FOR SOME INFINITELY

A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomi-als satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. Fo...

Infinitely Renormalizable Quadratic Polynomials

We prove that the Julia set of a quadratic polynomial which admits an infinite sequence of unbranched, simple renormalizations with complex bounds is locally connected. The method in this study is three-dimensional puzzles.