# Topology of irrationally indifferent attractors

## TL;DR

This paper classifies the topological structure of post-critical sets in certain holomorphic systems with irrationally indifferent fixed points, revealing a trichotomy based on the rotation number's arithmetic properties.

## Contribution

It establishes a trichotomy for the topology of post-critical sets in holomorphic systems, linking it to the arithmetic of the rotation number at the fixed point.

## Key findings

- Post-critical sets are either Jordan curves, hairy Jordan curves, or Cantor bouquets.
- The topology depends on the arithmetic nature of the rotation number.
- This explains how invariant curves degenerate within Siegel disks.

## Abstract

We study the post-critical set of a class of holomorphic systems with an irrationally indifferent fixed point. We prove a trichotomy for the topology of the post-critical set based on the arithmetic of the rotation number at the fixed point. The only options are Jordan curves, a one-sided hairy Jordan curves, and Cantor bouquet. This explains the degeneration of the closed invariant curves inside the Siegel disks, as one varies the rotation number.