  An ergodic study of Painleve VI is developed. The chaotic nature of its
Poincare 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 Painleve VI, a
Riemann-Hilbert correspondence, the dynamical system of a birational map on a
cubic surface, and the Lefschetz fixed point formula.
