Rigidity of bounded-type Siegel polynomials

Abstract

In this paper, we study rigidity of polynomials of arbitrary degree in the presence of neutral dynamics. Specifically, we focus on {non-renormalizable} (in the sense of Douady and Hubbard) complex polynomials of degree $d \geqslant 2$ that possess a Siegel disk of bounded type rotation number. We refer to such maps as {atomic Siegel polynomials of bounded type}. In this setting, our main results are: (A) Atomic Siegel polynomials of bounded type have locally connected Julia sets; (B) these Julia sets are quasiconformally rigid, i.e., they do not support invariant line fields; (C) any two combinatorially equivalent atomic Siegel polynomials of bounded type coincide up to an affine change of coordinates. In particular, item (C) verifies the notorious {Combinatorial Rigidity Conjecture} for atomic Siegel polynomials of bounded type and arbitrary degree. By bringing neutral Siegel dynamics into the picture, we extend the celebrated higher-degree rigidity theorems of Avila--Kahn--Lyubich--Shen and Kozlovski--van Strien, which until now applied only in the Yoccoz setting, i.e., for finitely many times renormalizable polynomials without irrationally indifferent periodic points.