Rational maps with smooth degenerate Herman rings

TL;DR

This paper demonstrates the existence of rational maps with smooth degenerate Herman rings, using advanced quasiconformal surgery techniques, and also constructs maps with positive area Julia sets lacking certain dynamical features.

Contribution

It establishes the existence of rational maps with smooth degenerate Herman rings and constructs maps with positive area Julia sets without irrationally indifferent points.

Findings

Existence of rational maps with smooth degenerate Herman rings.

Construction of rational maps with positive area Julia sets and no Herman rings.

Continuity of surgery and control of Julia set area are key techniques.

Abstract

We prove the existence of rational maps having smooth degenerate Herman rings. This answers a question of Eremenko affirmatively. The proof is based on the construction of smooth Siegel disks by Avila, Buff and Ch\'{e}ritat as well as the classical Siegel-to-Herman quasiconformal surgery. A crucial ingredient in the proof is the surgery's continuity, which relies on the control of the loss of the area of quadratic filled-in Julia sets by Buff and Ch\'{e}ritat. As a by-product, we prove the existence of rational maps having a nowhere dense Julia set of positive area for which these maps have no irrationally indifferent periodic points, no Herman rings, and are not renormalizable.