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Bernoulli measure of complex admissible kneading sequences

Published online by Cambridge University Press:  11 March 2013

HENK BRUIN  and
DIERK SCHLEICHER
HENK BRUIN
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstraße 15, 1090 Vienna, Austria email [email protected]
DIERK SCHLEICHER
Affiliation:
Jacobs University Bremen, Research I, PO Box 750 561, D-28725 Bremen, Germany email [email protected]
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Abstract

Iterated quadratic polynomials give rise to a rich collection of different dynamical systems that are parametrized by a simple complex parameter $c$. The different dynamical features are encoded by the kneading sequence, which is an infinite sequence over $\{ \mathtt{0} , \mathtt{1} \} $. Not every such sequence actually occurs in complex dynamics. The set of admissible kneading sequences was described by Milnor and Thurston for real quadratic polynomials, and by the authors in the complex case. We prove that the set of admissible kneading sequences has positive Bernoulli measure within the set of sequences over $\{ \mathtt{0} , \mathtt{1} \} $.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 3 , June 2013 , pp. 821 - 830
Copyright
Copyright ©2013 Cambridge University Press

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References

Bruin, H. and Schleicher, D.. Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials. J. Lond. Math. Soc. 8 (2009), 502–522.Google Scholar
Bruin, H. and Schleicher, D.. Symbolic Dynamics of Quadratic Polynomials (Monograph), in preparation (see Report 7, Institute Mittag–Leffler, 2001/02).Google Scholar
Bruin, H., Kaffl, A. and Schleicher, D.. Existence of quadratic Hubbard trees. Fund. Math. 202 (2009), 251–279.Google Scholar
Douady, A. and Hubbard, J.. Études Dynamique des Polynômes Complexes I & II (The Orsay Notes). Publications Mathèmatiques d’Orsay, 1984–85, http://www.math.cornell.edu/~hubbard/OrsayEnglish.pdf.Google Scholar
Keller, K.. Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set (Lecture Notes in Mathematics, 1732). Springer, New York, 2000.Google Scholar
Lau, E. and Schleicher, D.. Internal addresses in the Mandelbrot set and irreducibility of polynomials. Stony Brook Preprint #19, 1994.Google Scholar
Milnor, J.. Dynamics in One Complex Variable. Introductory Lectures,(Annals of Mathematical Studies, 160). 3rd edn. Princeton University Press, Princeton, NJ, 2006.Google Scholar
Milnor, J. and Thurston, W.. On Iterated Maps of the Interval. (Lecture Notes in Mathematics, 1342). Ed. Alexander, J. C.. Springer, Heidelberg, 1988, pp. 465–563.Google Scholar
Penrose, C.. On quotients of shifts associated with dendrite Julia sets of quadratic polynomials. PhD Thesis, University of Coventry, 1994.Google Scholar
Poirier, A.. Hubbard trees. Fund. Math. 208 (2) (2010), 193–248.Google Scholar
Schleicher, D.. Internal addresses in the Mandelbrot set and Galois groups of polynomials. Preprint, arXiv:math/9411238.Google Scholar
Thurston, W.. On the geometry and dynamics of iterated rational maps. Complex Dynamics: Families and Friends. Ed. Schleicher, D.. AK Peters, Wellesley, 2009, pp. 3–109.Google Scholar