The topology and combinatorial structure of the Mandelbrot set (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence is realized. We use this procedure to find a large class of addresses that are nonrealizable, and to prove that all such trees in actually satisfy the Translation Principle (in contrast to ). We also study how the existence of a hyperbolic component with a given address depends on the degree d: addresses can be sorted into families so that at least one address of each family is realized for sufficiently large d.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm180-1-4,
author = {Virpi Kauko},
title = {Shadow trees of Mandelbrot sets},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {35-87},
zbl = {1047.37030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm180-1-4}
}
Virpi Kauko. Shadow trees of Mandelbrot sets. Fundamenta Mathematicae, Tome 177 (2003) pp. 35-87. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm180-1-4/