2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106393The number of periodic solutions to Painlevé VI along a Pochhammer loop is counted exactly. It is shown that the number grows exponentially with period, where the growth rate is determined explicitly. Principal ingredients of the computation 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.26 pages, 10 figures, 4 tablesAlgebraic GeometryDynamical Systems34M55; 37F10text