Arithmetic geometric model for the renormalisation of irrationally indifferent attractors

TL;DR

This paper develops a geometric and topological model for understanding the renormalisation and dynamics near irrationally indifferent fixed points in holomorphic maps, emphasizing the role of arithmetic properties of the rotation number.

Contribution

It introduces a novel geometric model that captures the arithmetic properties of the rotation number and applies it to describe the dynamics near irrationally indifferent fixed points.

Findings

Constructed a geometric model incorporating arithmetic properties.

Built a topological model for local dynamics.

Explained the topology and dynamics of the maximal invariant set.

Abstract

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points. The geometric model incorporates the fine arithmetic properties of the rotation number at the fixed point. Using this model for the renormalisation, we build a topological model for the dynamics of a holomorphic map near an irrationally indifferent fixed point. Then, we explain the topology of the maximal invariant set for the model, and also explain the dynamics of the map on the maximal invariant set.