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Markov extensions and lifting measures for complex polynomials

Published online by Cambridge University Press:  12 March 2007

HENK BRUIN  and
MIKE TODD
HENK BRUIN
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Guildford, Surrey GU2 7XH, UK (e-mail: [email protected], [email protected])
MIKE TODD
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Guildford, Surrey GU2 7XH, UK (e-mail: [email protected], [email protected])
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Abstract

For polynomials $f$ on the complex plane with a dendrite Julia set we study invariant probability measures, obtained from a reference measure. To do this we follow Keller [K1] in constructing canonical Markov extensions. We discuss ‘liftability’ of measures (both $f$-invariant and non-invariant) to the Markov extension, showing that invariant measures are liftable if and only if they have a positive Lyapunov exponent. We also show that $\delta$-conformal measure is liftable if and only if the set of points with positive Lyapunov exponent has positive measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 3 , June 2007 , pp. 743 - 768
Copyright
2007 Cambridge University Press

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