Holomorphic dynamics, Painlev\'e VI equation and Character Varieties

We study the monodromy of Painlev\'e VI equation from a dynamical point of view. This is applied to the description of bounded orbits, and to a proof of the irreducibility of Painlev\'e VI equation in the sens of Casale and Malgrange. On our way, we compute the entropy of each element of the monodromy group, and we precise the dictionary between character varieties and Painlev\'e equations

The fifty-two icosahedral solutions to Painleve VI

The solutions of the (nonlinear) Painleve VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F4 Weyl group action and many properties of the solutions will be given. There are 52 classes, the first ten of which correspond directly to the ten icosahedral entries on Schwarz's list of algebraic solutions of the hypergeometric equation. The next nine solutions are simple deformations of known PVI solutions (and have less than five branches) and five of the larger solutions are already known, due to work of Dubrovin and Mazzocco and Kitaev. Of the remaining 28 solutions we will find 20 explicitly using (the author's correction of) Jimbo's asymptotic formula. Amongst those constructed there is one solution that is 'generic' in that its parameters lie on none of the affine F4 hyperplanes, one that is equivalent to the Dubrovin--Mazzocco elliptic solution and three elliptic solutions that are related to the Valentiner three-dimensional complex reflection group, the largest having 24 branches.Comment: 28 pages, 2 tables, final version, to appear in Crelle's journal (minor corrections, added two solutions and remarked that the remaining 8 solutions may be obtained via quadratic transformations