Periodic Solutions to Painlev\'e VI and Dynamical System on Cubic Surface

The number of periodic solutions to Painlev\'e 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 Painlev\'e VI, a Riemann-Hilbert correspondence, the dynamical system of a birational map on a cubic surface, and the Lefschetz fixed point formula.