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arXiv · 2301.08597

Dynamics of the fifth Painlev{\'e} foliation

Emmanuel Paul·Jean-Pierre Ramis
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Abstract

The leaves of the Painlev{\'e} foliations appear as the isomonodromic deformations of a rank 2 linear connection on a moduli space of connections. Therefore they are the fibers of the Riemann-Hilbert correspondence that sends each connection on its monodromy data, and this correspondence induces a conjugation between the dynamics of the foliation and a dynamic on a space of representations of some fundamental groupoid (a character variety). This one can be identified to a family of cubic surfaces through trace coordinates. We describe here the dynamics on the character variety related to the Painlev{\'e} V equation. We have here to consider irregular connections, and the representations of wild groupoids. We describe and compare all the dynamics which appear on this wild character variety: the tame dynamics, the confluent dynamics, the canonical symplectic dynamics and the wild dynamics.

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Emmanuel Paul, Jean-Pierre Ramis. 2023-01-20. Dynamics of the fifth Painlev{\'e} foliation. https://arxiv.org/abs/2301.08597

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