An ergodic study of Painlevé VI is developed. The chaotic nature of its Poincaré return map is established for almost all loops. The exponential growth of the numbers of periodic solutions is also shown. Principal ingredients of the arguments are a moduli-theoretical formulation of Painlevé VI, a Riemann-Hilbert correspondence, the dynamical system of a birational map on a cubic surface, and the Lefschetz fixed point formula.

It is a basic problem to count the number of periodic points of a surface mapping, since the growth rate of this number as the period tends to infinity is an important dynamical invariant. However, this problem becomes difficult when the map admits curves of periodic points. In this situation we give a precise estimate of the number of isolated periodic points for an area-preserving birational map of a projective complex surface.

The aim of this paper is to construct many examples of rational surface automorphisms with positive entropy by means of the concept of orbit data. The concept enables us to introduce some mild and verifiable condition, and to show that if an orbit data satisfies the condition, then there exists an automorphism realizing the orbit data. Applying this result, we describe the set of entropy values of the rational surface automorphisms in terms of Weyl groups.

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We develop a new method for constructing K3 surfaces. We construct such a K3 surface X by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the K3 surface X is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction.