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.
The topology and combinatorial structure of the Mandelbrot set (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 . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence is realized....
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