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Transitive maps which are not ergodic with respect to Lebesgue measure

Published online by Cambridge University Press:  19 September 2008

Sebastian Van Strien
Sebastian Van Strien
Affiliation:
Mathematics Department, University of Amsterdam, Netherlands (e-mail: strien@fwi.uva.nl)
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Abstract

In this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996 , pp. 833 - 848
Copyright
Copyright © Cambridge University Press 1996

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