Shadow trees of Mandelbrot sets

Volume 180 / 2003

Virpi Kauko Fundamenta Mathematicae 180 (2003), 35-87 MSC: Primary 37F20; Secondary 37B10, 05C05. DOI: 10.4064/fm180-1-4


The topology and combinatorial structure of the Mandelbrot set ${\mathcal M}^d$ (of degree $d\ge 2$) 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 ${\mathcal M}^d$. Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, ${\mit \Lambda }^d$. 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 ${\mit \Lambda }^d$ actually satisfy the Translation Principle (in contrast to ${\mathcal M}^d$). 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 $d$.


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

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