Combinatorial rigidity for unicritical polynomials

By Artur Avila, Jeremy A. Kahn, M. Lyubich, W. Shen. 2005

54 citations recorded by the source corpus.

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. This implies that the connectedness locus (the "Multibrot set") is locally connected at the corresponding parameter values and generalizes Yoccoz's Theorem for quadratics to the higher degree case.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/bab07a742ef037f18a75c7433493fdcacb7409f5

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes (2010, background): “21They have been established, however, for the family of primitive real combinatorics [KL1], which covers all real combinatorics except period doubling. 22This result also follows from combination of [AKLS] (dealing with at most ﬁnitely renormalizable maps) and [Ch] (dealing with the inﬁnitely renor”
Local connectivity and quasi‐conformal rigidity of non‐renormalizable polynomials (2006, background): “) Our first result generalizes the celebrated result of [18] for finitely renormalizable quadratic maps, and of [1] for finitely renormalizable unicritical”
Proof of the Branner-Hubbard conjecture on Cantor Julia sets (2006, methodology): “It becomes a powerful tool in the study of dynamics of polynomials, see for example [2], [11], [12], [13], [16], [17], [18], [22], [23], [24], [25], [27], [35], and [37].”