Bounded Trajectories in the Sixth Painlevé Dynamics
Katsunori Iwasaki(Faculty of Mathematics, Kyushu University)
The sixth Painlevé equation PVI (κ) is a second-order nonlinear ordinary differential equation
with an independent variable x ∈ P1 − {0, 1, ∞} and an unknown function q = q(x),µ¶
µ¶
1 1
1
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112qxx =++q −++qx
2 q q−1 q−x xx x−1 q−x(1)½¾xx
−1x(x−1)q(q − 1)(q − x)
κ24 − κ21 2 + κ22+ (1 − κ23 ),+2x2 (x − 1)2
q(q − 1)2
(q − x)2
depending on complex parameters κ = (κ0 , κ1 , κ2 , κ3 , κ4 ) in a 4-dimensional affine space
K = {κ = (κ0 , κ1 , κ2 , κ3 , κ4 ) ∈ C5 : 2κ0 + κ1 + κ2 + κ3 + κ4 = 1}.

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Equation (1) is only a fragmentary appearance of a more substantial entity, called the sixth
Painlevé dynamics and still denoted by the same symbol PVI (κ), which is a time-dependent
Hamiltonian dynamical system on the fibrationπκ : M(κ) → T.

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Here the phase space M(κ) is the total space of a family of moduli spaces of certain stableparabolic connections on P1 with four regular singular points; the base space T is the configuration space of ordered four points in P1 , with t ∈ T playing the role of time variables; andπκ is the projection associating to each stable parabolic connection its ordered regular singularpoints. The fiber Mt (κ) = πκ−1 (t) over t ∈ T , which is referred to as the space of initial values
at time t, is the moduli space of stable parabolic connections with singularities fixed at t. Theparameters κ ∈ K represent data on the local exponents of stable parabolic connections.
The space of initial values Mt (κ) has the structure of a generalized Halphen surface in
Sakai’s terminology, or that of an Okamoto-Painlevé pair in Saito-Takebe-Terajima’s terminology. Namely, the moduli space Mt (κ) admits a natural compactification Mt (κ) with a uniqueeffective anti-canonical divisor Yt (κ) ∈ | − KMt (κ) | such that
Mt (κ) = Mt (κ) − Yt (κ)red ,(4)where Mt (κ) is known to be isomorphic to an 8-point blow-up of the Hirzebruch surface ofdegree 2. The divisor Yt (κ) is called the vertical leaf at time t, each point of which may bethought of as a point at infinity of the space Mt (κ).
The dynamical system PVI (κ) enjoys the Painlevé property, that is, any local trajectory canbe continued endlessly along any path on the time-variable space T to yield a global trajectory.
Thus one can speak of the Poincaré return map homomorphism, or the nonlinear monodromy,PRMκ : π1 (T, t) → Aut(Mt (κ)),(5)

which faithfully describes the multivaluedness of global trajectories. Then one can say that a
π1 (T, t)-orbit in Mt (κ) just represents a global trajectory of PVI (κ), while a point of the orbit
signifies an initial value of the corresponding global trajectory.
1Now we make a definition: A global trajectory is said to be bounded if the corresponding
π1 (T, t)-orbit in Mt (κ) is bounded away from the vertical leaf Yt (κ), namely, a bounded trajectory is such a trajectory that remains bounded, accumulating no points at infinity, under the
analytic continuation along any path in T . Then the following problem naturally occurs to us.
Problem. Determine all those parameters κ ∈ K for which PVI (κ) admits bounded trajectories.
Moreover, given any such parameter κ ∈ K, classify all the bounded trajectories of PVI (κ).
Being so difficult, at least to the speaker, this problem is far remote from its complete resolution.
In this talk, we make some preliminary discussions about this problem and some related issues.

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