An ergodic study of Painlevé VI

Abstract

An ergodic study of Painlevé VI is developed. The chaotic nature of its Poincaré 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 Painlevé VI, a Riemann-Hilbert correspondence, the dynamical system of a birational map on a cubic surface, and the Lefschetz fixed point formula.

Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation

“…This statement is obtained in [7] by a deformation argument: the topological entropy does not depend on the parameters (A, B, C, D); it suffices to compute it in the case of the Cayley cubic by looking at the birational action at infinity. This improves the previous work [23] where the authors provided an algorithm to compute the topological entropy for smooth cubics.…”

“…This classification is compatible with the description of mapping classes, Dehn twists corresponding to parabolic transformations, and pseudo-Anosov mappings to hyperbolic automorphisms. The most striking result in that direction is summarized in the following theorem (see [23,7]). This statement is obtained in [7] by a deformation argument: the topological entropy does not depend on the parameters (A, B, C, D); it suffices to compute it in the case of the Cayley cubic by looking at the birational action at infinity.…”

Algebraic solutions of the sixth Painlevé equation

“…Direct calculation shows that ρ is in fact a bijection. Moreover the same result holds true if we replace in (19) affine D 4 by the full affine F 4 action and quotient the set of all triples (ω X , ω Y , ω Z ) by K 4 ⋊ S 3 as described above.…”

Periodic points for area-preserving birational maps of surfaces

“…On the other hand, since C ∈ X II ( f n ) passes through x, the set (18). Since C ∈ X 1 ( f m ) = X 1 ( f −m ) passes through x, any irreducible component of the germ C x defines a prime element p ∈ Λ(( f −m ) * x ).…”

“…This example arises as a special case of a 4-parameter family of dynamical systems on cubic surfaces derived from the nonlinear monodromy of the sixth Painlevé equation via the Riemann-Hilbert correspondence [17,18]. Let S be the projective cubic surface in P 3 defined by the homogeneous cubic equation: A natural (complex) area-form on S is given by its Poincaré residue:…”