Arithmetic Geometric Model for the Renormalisation of Bi-critical Irrationally Indifferent Attractors

TL;DR

This paper develops a geometric and topological model for understanding the renormalisation and local dynamics of holomorphic maps with irrationally indifferent fixed points and two critical points, incorporating arithmetic and angular properties.

Contribution

It introduces a novel arithmetic geometric model for renormalisation of bi-critical irrationally indifferent fixed points in holomorphic maps.

Findings

Constructed a geometric model for renormalisation.

Built a topological model for local dynamics.

Analyzed the topology of the maximal invariant set.

Abstract

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points of holomorphic maps with two critical points. The model incorporates arithmetic properties of the rotation number at the fixed point, as well as the "angle" between the two critical points. Using this model for the renormalisation, we build a topological model for the local dynamics of such maps. We also explain the topology of the maximal invariant set for the model, and the dynamics of the map on the maximal invariant set.