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Combinatorial rigidity for unicritical polynomials

Artur Avila·Jeremy Kahn·Mikhail Lyubich·Weixiao Shen
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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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Artur Avila, Jeremy Kahn, Mikhail Lyubich, Weixiao Shen. 2005-07-12. Combinatorial rigidity for unicritical polynomials. https://arxiv.org/abs/math/0507240

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