Collet, Eckmann and Hölder
- Cite this article as:
- Graczyk, J. & Smirnov, S. Invent math (1998) 133: 69. doi:10.1007/s002220050239
We prove that Collet-Eckmann condition for rational functions, which requires exponential expansion only along the critical orbits, yields the Hölder regularity of Fatou components. This implies geometric regularity of Julia sets with non-hyperbolic and critically-recurrent dynamics. In particular, polynomial Collet-Eckmann Julia sets are locally connected if connected, and their Hausdorff dimension is strictly less than 2. The same is true for rational Collet-Eckmann Julia sets with at least one non-empty fully invariant Fatou component.