Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials

• Autores: Henk Bruin, Dierk Schleicher
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 78, Nº 2, 2008, págs. 502-522
• Idioma: inglés
• DOI: 10.1112/jlms/jdn033
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• Resumen

  • Hubbard trees are invariant trees connecting the points of the critical orbits of post-critically finite polynomials. Douady and Hubbard showed in the Orsay Notes that they encode all combinatorial properties of the Julia sets. For quadratic polynomials, one can describe the dynamics as a subshift on two symbols, and itinerary of the critical value is called the kneading sequence. Whereas every (pre)periodic sequence is realized by an abstract Hubbard tree (see the authors� preprint from 2007), not every such tree is realized by a quadratic polynomial. In this paper, we give an Admissibility Condition that describes precisely which sequences correspond to quadratic polynomials. We identify the occurrence of the so-called �evil branch points� as the sole obstruction to being realizable. We also show how to derive the properties of periodic (branch) points in the tree (their periods, relative positions, number of arms and whether they are evil or not) from the kneading sequence