Recurrent critical points and typical limit sets of rational maps

Abstract

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