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Trees of visible components in the Mandelbrot set

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Kauko V.

Fundamenta Mathematicae

2000

tom 164

nr 1

41-60

We discuss the tree structures of the sublimbs of the Mandelbrot set M, using internal addresses of hyperbolic components. We find a counterexample to a conjecture by Eike Lau and Dierk Schleicher concerning topological equivalence between different trees of visible components, and give a new proof to a theorem of theirs concerning the periods of hyperbolic components in various trees.

Shadow trees of Mandelbrot sets

100%

Kauko V.

Fundamenta Mathematicae

2003

tom 180

nr 1

35-87

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.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Kauko V.

1 2003

1 2000

Wyniki wyszukiwania

Trees of visible components in the Mandelbrot set

Shadow trees of Mandelbrot sets