COLLET, ECKMANN AND HOLDER

We prove that Collet-Eckmann condition for rational functions, which r equires exponential expansion only along the critical orbits, yields t he Holder regularity of Fatou components. This implies geometric regul arity of Julia sets with non-hyperbolic and critically-recurrent dynam ics. In particular, polynomial Collet-Eckmann Julia sets are locally c onnected if connected, and their Hausdorff dimension is strictly less than 2, The same is true for rational Collet-Eckmann Julia sets with a t least one non-empty fully invariant Fatou component.