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

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