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.
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Oblatum 22-III-1996 & 15-VII-1997
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Graczyk, J., Smirnov, S. Collet, Eckmann and Hölder. Invent math 133, 69–96 (1998). https://doi.org/10.1007/s002220050239
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DOI: https://doi.org/10.1007/s002220050239