Rigidity of J-rotational rational maps and critical quasicircle maps

Willie Rush Lim

Published 2023-08-14Version 1

We say that a periodic quasicircle of a rational map is a rotation quasicircle if the return map is conjugate to an irrational rotation, and a Herman quasicircle if additionally it is not contained in the closure of any rotation domain. In the first part of this paper, we study rational maps which are J-rotational of bounded type, that is, geometrically finite away from rotation quasicircles of bounded type. We show that the Julia set of such a map does not support any invariant line field, and additionally has zero Lebesgue measure in the absence of Herman quasicircles. As an application, every rational map that admits a single bounded type Herman quasicircle of the simplest configuration must be combinatorially rigid and it always arises as a genuine limit of degenerating Herman rings of the same rotation number. In the second part, we initiate the study of renormalization theory of critical quasicircle maps, i.e. analytic self homeomorphisms of a quasicircle with a single critical point. We prove $C^{1+\alpha}$ rigidity of critical quasicircle maps with bounded type rotation number and construct the corresponding renormalization horseshoe. Our results generalize classical results of McMullen, de Faria, and de Melo.