Global asymptotics of the sixth Painlev\'e equation in Okamoto's space

TL;DR

This paper investigates the behavior of solutions to the sixth Painlevé equation near zero, revealing the structure of their limit sets, the unbounded nature of their poles and zeros, and the dynamics in Okamoto's space.

Contribution

It provides a detailed analysis of the asymptotic dynamics of sixth Painlevé solutions in Okamoto's space, including the description of the repeller set and the properties of their limit sets.

Findings

The repeller set for solutions is characterized.

Number of poles and zeros is unbounded for general solutions.

Each solution's complex limit set is compact and connected.

Abstract

We study dynamics of solutions in the initial value space of the sixth Painlev\'e equation as the independent variable approaches zero. Our main results describe the repeller set, show that the number of poles and zeroes of general solutions is unbounded, and that the complex limit set of each solution exists and is compact and connected.