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Dynamics of the Fifth Painlevé Foliation

In: Handbook of Geometry and Topology of Singularities VI: Foliations

Author

Listed:
  • Emmanuel Paul

    (Institute of Mathematics of Toulouse)

  • Jean-Pierre Ramis

    (Institut de Mathématiques de Toulouse (IMT)
    Institut de France (Académie des Sciences))

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.

Suggested Citation

  • Emmanuel Paul & Jean-Pierre Ramis, 2024. "Dynamics of the Fifth Painlevé Foliation," Springer Books, in: Felipe Cano & José Luis Cisneros-Molina & Lê Dũng Tráng & José Seade (ed.), Handbook of Geometry and Topology of Singularities VI: Foliations, chapter 0, pages 307-381, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-54172-8_9
    DOI: 10.1007/978-3-031-54172-8_9
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