Recurrent critical points and typical limit sets of rational maps

Abstract:

We consider a rational map $f:\widehat {\mathbb {C}}\to \widehat {\mathbb {C}}$ of the Riemann sphere with normalized Lebesgue measure $\mu$ and show that if there is a subset of the Julia set $J(f)$ of positive $\mu$-measure whose points have limit sets not contained in the union of the limit sets of recurrent critical points, then $\omega (x)=\widehat {\mathbb {C}}=J(f)$ for $\mu$-a.e. point $x$ and $f$ is conservative, ergodic and exact.