Dynamics of the Fifth Painlevé Foliation

Emmanuel Paul () and Jean-Pierre Ramis ()
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Emmanuel Paul: Institute of Mathematics of Toulouse
Jean-Pierre Ramis: Institut de Mathématiques de Toulouse (IMT)

Chapter Chapter 9 in Handbook of Geometry and Topology of Singularities VI: Foliations, 2024, pp 307-381 from Springer

Abstract: Abstract 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 irreducible components of 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 dynamics 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.

Date: 2024
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DOI: 10.1007/978-3-031-54172-8_9

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