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## Fundamenta Mathematicae

2003 | 180 | 1 | 35-87
Tytuł artykułu

### Shadow trees of Mandelbrot sets

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EN
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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.
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Tom
Numer
Strony
35-87
Opis fizyczny
Daty
wydano
2003
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autor
• Department of Mathematics, P.O. Box 35, FIN-40014 University of Jyväskylä, Finland
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