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Markov extensions and lifting measures for complex polynomials

Henk Bruin·Mike Todd
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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 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 $δ$-conformal measure is liftable if and only if the set of points with positive Lyapunov exponent has positive measure.

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Henk Bruin, Mike Todd. 2006-10-16. Markov extensions and lifting measures for complex polynomials. https://arxiv.org/abs/math/0507543

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