Existence of quadratic Hubbard trees

Henk Bruin ¹   ; Alexandra Kaffl ²   ; Dierk Schleicher ²

¹ Department of Mathematics University of Surrey Guildford GU2 7XH, United Kingdom
² School of Engineering and Science Jacobs University Bremen P.O. Box 750 561 D-28725 Bremen, Germany

Fundamenta Mathematicae, Tome 202 (2009) no. 3, pp. 251-279

Résumé

A (quadratic) Hubbard tree is an invariant tree connecting the critical orbit within the Julia set of a postcritically finite (quadratic) polynomial. It is easy to read off the kneading sequences from a quadratic Hubbard tree; the result in this paper handles the converse direction.Not every sequence on two symbols is realized as the kneading sequence of a real or complex quadratic polynomial. Milnor and Thurston classified all real-admissible sequences, and we give a classification of all complex-admissible sequences in \cite{BS}. In order to do this, we show here that every periodic or preperiodic sequence is realized by a unique abstract Hubbard tree. Real or complex admissibility of the sequence depends on whether this abstract tree can be embedded into the real line or complex plane so that the dynamics preserves the embedded, and this can be studied in terms of branch points of the abstract Hubbard tree.

Détail

DOI : 10.4064/fm202-3-4

Keywords: quadratic hubbard tree invariant tree connecting critical orbit within julia set postcritically finite quadratic polynomial easy read off kneading sequences quadratic hubbard tree result paper handles converse direction every sequence symbols realized kneading sequence real complex quadratic polynomial milnor thurston classified real admissible sequences classification complex admissible sequences cite order here every periodic preperiodic sequence realized unique abstract hubbard tree real complex admissibility sequence depends whether abstract tree embedded real line complex plane dynamics preserves embedded studied terms branch points abstract hubbard tree