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...

Physical Measures for Infinitely Renormalizable Lorenz Maps

A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics. Namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article we construct examples of i...

Renormalization on One-dimensional Folding Maps

Some techniques and results in the renormalization theory of real and complex dynamical systems are summarized. The construction of the induced Markov map of [−1, 1] from a Feigenbaum-like map is presented. We show that this induced Markov map has bounded geometry. We discuss some property of infinitely renormalizable quadratic polynomials and show that the Julia set of an infinitely renormaliz...