Shadow trees of Mandelbrot sets

Virpi Kauko. "Shadow trees of Mandelbrot sets." Fundamenta Mathematicae 180.1 (2003): 35-87. <http://eudml.org/doc/282662>.

@article{VirpiKauko2003,
abstract = {The topology and combinatorial structure of the Mandelbrot set $ℳ ^\{d\}$ (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 $ℳ ^\{d\}$. Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, $Λ^\{d\}$. 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 $Λ^\{d\}$ actually satisfy the Translation Principle (in contrast to $ℳ ^\{d\}$). 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.},
author = {Virpi Kauko},
journal = {Fundamenta Mathematicae},
keywords = {topology and combinatorial structure; Mandelbrot set; symbolic dynamics},
language = {eng},
number = {1},
pages = {35-87},
title = {Shadow trees of Mandelbrot sets},
url = {http://eudml.org/doc/282662},
volume = {180},
year = {2003},
}

TY - JOUR
AU - Virpi Kauko
TI - Shadow trees of Mandelbrot sets
JO - Fundamenta Mathematicae
PY - 2003
VL - 180
IS - 1
SP - 35
EP - 87
AB - The topology and combinatorial structure of the Mandelbrot set $ℳ ^{d}$ (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 $ℳ ^{d}$. Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, $Λ^{d}$. 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 $Λ^{d}$ actually satisfy the Translation Principle (in contrast to $ℳ ^{d}$). 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.
LA - eng
KW - topology and combinatorial structure; Mandelbrot set; symbolic dynamics
UR - http://eudml.org/doc/282662
ER -