Degeneration of the Julia set to singular loci of algebraic curves

TL;DR

This paper investigates how Julia sets of rational maps evolve as the maps transition from non-integrable to integrable, revealing that Julia sets approach indeterminate points along algebraic curves which are singular loci.

Contribution

It demonstrates that Julia sets tend to approach indeterminate points along algebraic curves that are singular loci during the transition from non-integrable to integrable maps.

Findings

Julia sets approach indeterminate points along algebraic curves.

Indeterminate points are identified as singular loci of these curves.

The behavior is analyzed during the continuous transition from non-integrable to integrable maps.

Abstract

We show that, when a non-integrable rational map changes to an integrable one continuously, a large part of the Julia set of the map approach indeterminate points (IDP) of the map along algebraic curves. We will see that the IDPs are singular loci of the curves.