Ergodic Theory and Dynamical Systems



Markov extensions and lifting measures for complex polynomials


HENK BRUIN a1 and MIKE TODD a1
a1 Department of Mathematics, University of Surrey, Guildford, Guildford, Surrey GU2 7XH, UK (e-mail: h.bruin@surrey.ac.uk, m.todd@surrey.ac.uk)

Article author query
bruin h   [Google Scholar] 
todd m   [Google Scholar] 
 

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.

(Published Online March 12 2007)
(Received July 26 2005)
(Revised September 26 2006)