Institute for Mathematical Sciences

Preprint ims00-09

J. Rivera-Letelier
Rational maps with decay of geometry: rigidity, Thurston's algorithm and local connectivity.

Abstract: We study dynamics of rational maps that satisfy a decay of geometry condition. Well known conditions of non-uniform hyperbolicity, like summability condition with exponent one, imply this condition. We prove that Julia sets have zero Lebesgue measure, when not equal to the whole sphere, and in the polynomial case every connected component of the Julia set is locally connected. We show how rigidity properties of quasi-conformal maps that are conformal in a big dynamically defined part of the sphere, apply to dynamics. For example we give a partial answer to a problem posed by Milnor about Thurston's algorithm and we give a proof that the Mandelbrot set, and its higher degree analogues, are locally connected at parameters that satisfy the decay of geometry condition. Moreover we prove a theorem about similarities between the Mandelbrot set and Julia sets. In an appendix we prove a rigidity property that extends a key situation encountered by Yoccoz in his proof of local connectivity of the Mandelbrot set at at most finitely renormalizable parameters.
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