Dynamics of the Sixth Painlev\'e Equation

TL;DR

This paper explores the complex geometric and dynamical properties of the sixth Painlevé equation using algebraic geometry and Riemann-Hilbert techniques, extending some results to Garnier systems.

Contribution

It provides a comprehensive analysis of the sixth Painlevé equation's dynamics through a novel geometric approach, connecting it with Garnier systems.

Findings

Revealed rich geometric structures underlying the sixth Painlevé equation

Extended techniques to Garnier systems

Enhanced understanding of the equation's dynamical nature

Abstract

The sixth Painlev\'e equation is hiding extremely rich geometric structures behind its outward appearance. This article tries to give as a total picture as possible of its dynamical natures, based on the Riemann-Hilbert approach recently developed by the authors, using various techniques from algebraic geometry. A good part of the contents is extended to Garnier systems, while this article is restricted to the original sixth Painlev\'e equation.