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The article studies the Fifth Painlev\'e equation and of the nonlinear Stokesphenomenon at its irregular singularity at infinity from the point of view ofconfluence from the Sixth Painlev\'e equation. This approach is developpedseparately on both sides of the Riemann-Hilbert correspondance. On the side ofthe nonlinear Painlev\'e-Okamoto foliation the relation between the nonlinearmonodromy 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 indetail. 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 transformationfrom the character variety (the space of the linear monodromy data) associatedto Painlev\'e VI. This allows to transport the known description of the actionof the nonlinear monodromy of Painlev\'e VI on its character variety to that ofPainlev\'e V, and to provide explicit formulas for the action of the "nonlinearwild monodromy" of Painlev\'e V on its character variety.

The wild monodromy of the Fifth Painlev\'e equation and its action on wild character variety: an approach of confluence