  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.
