Combinatorial rigidity for unicritical polynomials

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.

LaTeX, 12 pages

References in corpus (4)

• Rational parameter rays of the Mandelbrot set
• Holomorphic Removability of Julia Sets
• The Quasi-Additivity Law in Conformal Geometry
• Local connectivity of Julia sets for unicritical polynomials

Cited by in corpus (3)

• A priori bounds for some infinitely renormalizable quadratics: III. Molecules
• A priori bounds for some infinitely renormalizable quadratics: II. Decorations
• Combinatorial rigidity for some infinitely renormalizable unicritical polynomials