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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 henk.bruin@univie.ac.at
DIERK SCHLEICHER
Affiliation:
Jacobs University Bremen, Research I, PO Box 750 561, D-28725 Bremen, Germany email dierk@jacobs-university.de
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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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