Isomonodromy equations, algebraic solutions and dynamics

Abstract : We call isomonodromic deformation any family of logarithmic flat connections over a punctured sphere having the same monodromy representation up to global conjugacy. These objects are parametrised by the solutions of a particular family of partial differential equations called Garnier systems, which are equivalent to the Painlevé VI equations in the four punctured case. The purpose of this thesis is to construct new algebraic solutions of these systems in the five punctured case. First, we give a classification of algebraic isomonodromic deformations obtained by restricting to lines some logarithmic flat connection over the complex projective plane whose singular locus is a quintic curve. We obtain two new families of algebraic solutions of the associated Garnier system. In a second part, we use the fact that any algebraic isomonodromic deformation corresponds to a finite orbit under the mapping class group action on the character variety of the five punctured sphere to obtain new examples of such orbits. We do this by using Katz's middle convolution on representations of free groups. Finally, we give a partial generalisation of this procedure in the case of a twice punctured complex torus.
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https://tel.archives-ouvertes.fr/tel-01368560
Contributor : Arnaud Girand <>
Submitted on : Tuesday, October 25, 2016 - 5:00:26 PM
Last modification on : Thursday, November 15, 2018 - 11:56:46 AM

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  • HAL Id : tel-01368560, version 2

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Arnaud Girand. Isomonodromy equations, algebraic solutions and dynamics. Complex Variables [math.CV]. Université de Rennes 1; Université Bretagne Loire, 2016. English. ⟨tel-01368560v2⟩

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