# Hairy Cantor sets

## TL;DR

The paper introduces hairy Cantor sets, a new topological object with universal features, and proves their ambient homeomorphism in the plane, linking complex dynamics and renormalisation theory.

## Contribution

It provides an axiomatic characterization of hairy Cantor sets and establishes their topological equivalence in the plane, connecting dynamics of holomorphic maps with arithmetic conditions.

## Key findings

- Hairy Cantor sets share universal topological features.
- Any two hairy Cantor sets in the plane are ambiently homeomorphic.
- They link polynomial-like renormalisation with circle diffeomorphism dynamics.

## Abstract

We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeomorphic. Hairy Cantor sets appear in the study of the dynamics of holomorphic maps with infinitely many renormalisation structures. They are employed to link the fundamental concepts of polynomial-like renormalisation by Douady-Hubbard with the arithmetic conditions obtained by Herman-Yoccoz in the study of the dynamics of analytic circle diffeomorphisms.