Ergodicity of conformal measures for unimodal polynomials

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