Rigidity of bounded-type Siegel polynomials

Abstract

We establish rigidity for a class of higher-degree complex polynomials with irrationally indifferent dynamics. Specifically, we consider non-renormalizable (in the sense of Douady and Hubbard) polynomials of degree $d\geqslant 2$ with a Siegel disk whose rotation number is of bounded type. We call such maps atomic Siegel polynomials of bounded type. Our main results are: (A) The Julia set of every atomic Siegel polynomial of bounded type is locally connected; (B) Every atomic Siegel polynomial of bounded type is quasiconformally rigid; equivalently, its Julia set supports no invariant line fields; (C) Any two combinatorially equivalent atomic Siegel polynomials of bounded type are affinely conjugate. In particular, (C) proves the Combinatorial Rigidity Conjecture for atomic Siegel polynomials of bounded type in arbitrary degree. This extends the higher-degree rigidity theory of Avila--Kahn--Lyubich--Shen and Kozlovski--van Strien to the setting of irrationally indifferent dynamics, a setting not previously covered by Yoccoz-type rigidity results.