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Measure of the Julia set of the Feigenbaum map with infinite criticality

Published online by Cambridge University Press:  29 June 2009

GENADI LEVIN  and
GRZEGORZ ŚWIA̧TEK
GENADI LEVIN
Affiliation:
Department of Mathematics, Hebrew University, Jerusalem 91904, Israel (email: [email protected])
GRZEGORZ ŚWIA̧TEK
Affiliation:
Wydział MiNI, Politechnika Warszawska, Plac Politechniki 1, 00-661 Warszawa, Poland (email: [email protected])
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Abstract

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets goes to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 3 , June 2010 , pp. 855 - 875
Copyright
Copyright © Cambridge University Press 2009

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