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Journal of the London Mathematical Society Advance Access originally published online on July 10, 2008
Journal of the London Mathematical Society 2008 78(2):502-522; doi:10.1112/jlms/jdn033
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© 2008 London Mathematical Society

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

Henk Bruin

Department of Mathematics
University of Surrey
Guildford
GU2 7XH
United Kingdom

Dierk Schleicher

School of Engineering and Science
Jacobs University Bremen
P.O. Box 750 561
D-28725 Bremen
Germany
dierk@jacobs-university.de

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.

2000 Mathematics Subject Classification 37F20 (primary), 37B10, 37E25 (secondary).

Received January 26, 2008; published online July 10, 2008.


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