Combinatorial rigidity for some infinitely renormalizable unicritical polynomials

Cheraghi, Davoud

doi:10.1090/S1088-4173-2010-00216-X

Combinatorial rigidity for some infinitely renormalizable unicritical polynomials
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Conform. Geom. Dyn. 14 (2010), 219-255 Request permission

Abstract:

Here we prove that infinitely renormalizable unicritical polynomials $P_c:z \mapsto z^d+c$, with $c\in \mathbb {C}$, satisfying a priori bounds and a certain “combinatorial” condition are combinatorially rigid. This implies the local connectivity of the connectedness loci (the Mandelbrot set when $d=2$) at the corresponding parameters.

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