Existence of quadratic Hubbard trees
Volume 202 / 2009
Abstract
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.
