Singular cubic surfaces and the dynamics of Painleve VI

Abstract

We develop a dynamical study of the sixth Painleve equation for all parameters generalizing an earlier work for generic parameters. Here the main focus of this paper is on non-generic parameters, for which the corresponding character variety becomes a cubic surface with simple singularities and the Riemann-Hilbert correspondence is a minimal resolution of the singular surface, not a biholomorphism as in the generic case. Introducing a suitable stratification on the parameter space and based on geometry of singular cubic surfaces, we establish a chaotic nature of the nonlinear monodromy map of Painleve VI and give a precise estimate for the number of its isolated periodic solutions.