Dynamics of the Sixth Painlevé Equation

Abstract

The sixth Painlevé 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 Painlevé equation.