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Summary: COMBINATORIAL RIGIDITY
FOR UNICRITICAL POLYNOMIALS
ARTUR AVILA, JEREMY KAHN, MIKHAIL LYUBICH AND WEIXIAO SHEN
Abstract. We prove that any unicritical polynomial fc : z zd + 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.
1. Introduction
Let us consider the one-parameter family of unicritical polynomials
fc : z zd
+ c, c C,
of degree d 2. Let M = Md = {c C, the Julia set of fc is connected} be the
connectedness locus of this family. In the case of quadratic polynomials (d = 2),
it is known as the Mandelbrot set, while in the higher degree case it is sometimes
called the Multibrot set (see [Sc2]).
Rigidity is one of the most remarkable phenomena observed in holomorphic dy-
namics. In the unicritical case this phenomenon assumes (conjecturally) a par-
ticularly strong form of combinatorial rigidity: combinatorially equivalent non-
hyperbolic maps are conformally equivalent. This Rigidity Conjecture is equiv-
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