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Combinatorial rigidity for unicritical polynomials. (English) Zbl 1204.37047
For a positive integer \(d \geq 2\) consider the one-parameter family of unicritical polynomials \(f_c\) with \(f_c(z):=z^d+c\), where \(c \in \mathbb C\). The connectedness locus \({\mathcal M}_d\) of this familiy is the set of parameters \(c\) such that the corresponding Julia set is connected. The set \({\mathcal M}_2\) is the well-known Mandelbrot set while for higher degrees these sets are also called Multibrot sets. The famous MLC conjecture asserts that the Mandelbrot set is locally connected. This is equivalent to the rigidity conjecture, that is that combinatorially equivalent nonhyperbolic maps are conformally equivalent.
About 15 years ago Yoccoz proved that the Mandelbrot set is locally connected at all nonhyperbolic parameter values which are at most finitely renormalizable. In this paper the authors prove that a unicritical polynomial which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. This implies that the corresponding Multibrot set is locally connected at the corresponding parameter values and generalizes Yoccoz’s theorem for quadratic polynomials to the higher degree case.

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
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