Periodic Solutions to Painlevé VI and Dynamical System on Cubic Surface
arXiv:math/0512583
Abstract
The 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 tables