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Conformal Geometry and Dynamics

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Combinatorial rigidity for some infinitely renormalizable unicritical polynomials

Author: Davoud Cheraghi
Journal: Conform. Geom. Dyn. 14 (2010), 219-255
MSC (2010): Primary 37F45; Secondary 37F25, 37F30
Published electronically: September 15, 2010
MathSciNet review: 2719786
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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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Additional Information

Davoud Cheraghi
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
Address at time of publication: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received by editor(s): April 20, 2008
Received by editor(s) in revised form: December 28, 2009, and June 2, 2010
Published electronically: September 15, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.