Markov extensions and lifting measures for complex polynomials

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 δ-conformal measure is liftable if and only if the set of points with positive Lyapunov exponent has positive measure.