Shadow trees of Mandelbrot sets

Virpi Kauko¹

¹ Department of Mathematics P.O. Box 35 FIN-40014 University of Jyväskylä, Finland

Fundamenta Mathematicae, Tome 180 (2003) no. 1, pp. 35-87.

Résumé

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 $d$.

Détail

DOI : 10.4064/fm180-1-4

Keywords: topology combinatorial structure mandelbrot set mathcal degree studied using symbolic dynamics each parameter mapped kneading sequence equivalently internal address every sequence realized parameter mathcal abstract mandelbrot set subspace larger partially ordered symbol space mit lambda paper algorithm construct visible trees symbolic sequences which works whether sequence realized procedure large class addresses nonrealizable prove trees mit lambda actually satisfy translation principle contrast mathcal study existence hyperbolic component given address depends degree addresses sorted families least address each family realized sufficiently large