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Inventiones mathematicae

, Volume 133, Issue 1, pp 69–96 | Cite as

Collet, Eckmann and Hölder

  • Jacek Graczyk
  • Stas Smirnov
Article

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.

Keywords

Rational Function Hausdorff Dimension Critical Orbit Exponential Expansion Fatou Component 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Jacek Graczyk
    • 1
  • Stas Smirnov
    • 1
  1. 1.Department of Mathematics, Caltech, Pasadena, CA 91125, USAUS

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