Inventiones mathematicae

, Volume 133, Issue 1, pp 69–96

Collet, Eckmann and Hölder

Authors

  • Jacek Graczyk
    • Department of Mathematics, Caltech, Pasadena, CA 91125, USA
  • Stas Smirnov
    • Department of Mathematics, Caltech, Pasadena, CA 91125, USA
Article

DOI: 10.1007/s002220050239

Cite this article as:
Graczyk, J. & Smirnov, S. Invent math (1998) 133: 69. doi:10.1007/s002220050239

Abstract.

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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998