The wild monodromy of the Fifth Painlevé equation and its action on wild character variety: an approach of confluence

Abstract: The article studies the Fifth Painlev\'e equation and of the nonlinear Stokes phenomenon at its irregular singularity at infinity from the point of view of confluence from the Sixth Painlev\'e equation. This approach is developped separately on both sides of the Riemann-Hilbert correspondance. On the side of the nonlinear Painlev\'e-Okamoto foliation the relation between the nonlinear monodromy group of Painlev\'e VI and the "nonlinear wild monodromy pseudogroup" of Painlev\'e V (that is the pseudogroup generated by nonlinear monodromy, nonlinear Stokes operators and nonlinear exponential torus) is explained in detail. On the side of the corresponding linear isomonodromic problem, the "wild" character variety (the space of the linear monodromy and Stokes data) associated to Painlev\'e V is constructed through a birational transformation from the character variety (the space of the linear monodromy data) associated to Painlev\'e VI. This allows to transport the known description of the action of the nonlinear monodromy of Painlev\'e VI on its character variety to that of Painlev\'e V, and to provide explicit formulas for the action of the "nonlinear wild monodromy" of Painlev\'e V on its character variety.