Institute for Mathematical Sciences

Preprint ims05-05

Artur Avila, Jeremy Kahn, Mikhail Lyubich and Weixiao Shen
Combinatorial rigidity for unicritical polynomials

Abstract: We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the ``Multibrot set'') is locally connected at the corresponding parameter values. It generalizes Yoccoz's Theorem for quadratics to the higher degree case.
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