Shadow trees of Mandelbrot sets

• Autorzy: Virpi Kauko
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 05C05: Trees
• 37F20: Combinatorics and topology
• 37B10: Symbolic dynamics

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2003
• Tom: 180
• Numer: 1
• Strony: 35-87

Daty

wydano

2003

Twórcy

autor

Virpi Kauko

• Department of Mathematics, P.O. Box 35, FIN-40014 University of Jyväskylä, Finland

Identyfikatory

DOI

10.4064/fm180-1-4