Smooth Combs Inside Hedgehogs

TL;DR

This paper constructs smooth combs within hedgehogs using advanced Riemann surface techniques, revealing new geometric structures in complex dynamics related to non-linearisable holomorphic maps with indifferent fixed points.

Contribution

It introduces a novel method to embed smooth combs inside hedgehogs associated with commuting non-linearisable holomorphic maps, expanding understanding of their geometric properties.

Findings

Hedgehogs contain smooth $C^{ abla}$ combs constructed via Riemann surface techniques.

The combs are made of smooth curves with transversally bi-Hölder regularity.

The work links complex dynamics with geometric structures in Riemann surfaces.

Abstract

We use techniques of tube-log Riemann surfaces due to R.Perez-Marco to construct a hedgehog containing smooth $C^{\infty}$ combs. The hedgehog is a common hedgehog for a family of commuting non-linearisable holomorphic maps with a common indifferent fixed point. The comb is made up of smooth curves, and is transversally bi-H\"older regular.