Markov extensions and lifting measures for complex polynomials
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.