Rigidity of bounded-type Siegel polynomials

TL;DR

This paper proves that non-renormalizable degree d complex polynomials with a bounded type Siegel disk have locally connected Julia sets, are quasiconformally rigid, and satisfy the combinatorial rigidity conjecture, extending previous higher-degree rigidity results.

Contribution

It establishes the rigidity and local connectivity of Julia sets for atomic Siegel polynomials of bounded type, confirming the combinatorial rigidity conjecture in this setting.

Findings

Julia sets are locally connected

Julia sets are quasiconformally rigid

Combinatorial equivalence implies affine conjugacy

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.