2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/111710An 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.40 pages, 11 figures, 4 tables, 32 references, an upgraded version of the article: arXiv: math.AG/0512583Algebraic GeometryDynamical Systems34M55; 37F10text