Chapitre D'ouvrage Année : 2024

Dynamics of the fifth Painlevé foliation

Résumé

The leaves of the Painlevé 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 Painlevé 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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Dates et versions

hal-03941706 , version 1 (16-01-2023)

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Emmanuel Paul, Jean-Pierre Ramis. Dynamics of the fifth Painlevé foliation. Handbook of geometry and topology of singularities VI: foliations, Springer, pp 307-381, 2024, ⟨10.1007/978-3-031-54172-8_9⟩. ⟨hal-03941706⟩
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