Institute for Mathematical Sciences
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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