Ergodicity of conformal measures for unimodal polynomials

Abstract

Let f f be a polynomial and μ \mu a conformal measure for f f , i.e., a Borel probability measure μ \mu with Jacobian equal to | D f ( z ) | δ |Df(z)|^{\delta } . We show that if f f is a real unimodal polynomial (a polynomial with just one critical point), then μ \mu is ergodic. We also show that μ \mu is ergodic if f f is a complex unimodal polynomial with one parabolic periodic point or a quadratic polynomial in the S L \mathcal {SL} class with a priori bounds (as defined in Lyubich (1997)).