## 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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