Holomorphic dynamics, Painlev\'e VI equation and Character Varieties

We study the monodromy of Painlev\'e VI equation from a dynamical point of view. This is applied to the description of bounded orbits, and to a proof of the irreducibility of Painlev\'e VI equation in the sens of Casale and Malgrange. On our way, we compute the entropy of each element of the monodromy group, and we precise the dictionary between character varieties and Painlev\'e equations

Ramified covers and tame isomonodromic solutions on curves

In this paper, we investigate the possibility of constructing isomonodromic deformations of logarithmic connections on curves by using ramified covers. We give new examples and prove a classification result

The holonomy group at infinity of the Painleve VI Equation

We prove that the holonomy group at infinity of the Painleve VI equation is virtually commutative.Comment: 21 pages, 3 figures, Added references, Corrected typo

A new two--parameter family of isomonodromic deformations over the five punctured sphere

The object of this paper is to describe an explicit two--parameter family of logarithmic flat connections over the complex projective plane. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a circle and three tangent lines. By restricting them to generic lines we get an algebraic family of isomonodromic deformations of the five--punctured sphere. This yields new algebraic solutions of a Garnier system. Finally, we use the associated Riccati one--forms to construct an interesting non--generic family of transversally projective Lotka--Volterra foliations.Comment: English text, 30 page

Tau functions as Widom constants

We define a tau function for a generic Riemann-Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices. Its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the Riemann-Hilbert problem. In the case of one circle, the tau function coincides with Widom's determinant arising in the asymptotics of block Toeplitz matrices. Our construction gives the Jimbo-Miwa-Ueno tau function for Riemann-Hilbert problems of isomonodromic origin (Painlev\'e VI, V, III, Garnier system, etc) and the Sato-Segal-Wilson tau function for integrable hierarchies such as Gelfand-Dickey and Drinfeld-Sokolov.Comment: 26 pages, 6 figure

Monodromy dependence and connection formulae for isomonodromic tau functions

We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for generic Painlev\'e VI tau function. The result proves the conjectural formula for this constant proposed in \cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlev\'e II tau function.Comment: 54 pages, 6 figures; v4: rewritten Introduction and Subsection 3.3, added few refs to match published articl

Studies on the geometry of Painlevé equations

This thesis is a collection of work within the geometric framework for Painlevé equations. This approach was initiated by the Japanese school, and is based on studying Painlevé equations (differential or discrete) via certain rational surfaces associated with affine root systems. Our work is grouped into two main themes: on the one hand making use of tools and techniques from the geometric framework to study problems from applications where Painlevé equations appear, and on the other hand developing and extending the geometric framework itself. Differential and discrete Painlevé equations arise in a wide range of areas of mathematics and physics, and we present a general procedure for solving the identification problem for Painlevé equations. That is, if a differential or discrete system is suspected to be equivalent to one of Painlevé type, we outline a method, based on constructing the associated surfaces explicitly, for identifying the system with a standard example, in which case known results can be used, and demonstrate it in the case of equations appearing in the theory of orthogonal polynomials. Our results on the geometric framework itself begin with an observation of a new class of discrete equations that can described through the geometric theory, beyond those originally defined by Sakai in terms of translation symmetries of families of surfaces. To be precise, we build on previous studies of equations corresponding to non-translation symmetries of infinite order (so-called projective reductions, with fewer parameters than translations of the same surface type) and show that Sakai’s theory allows for integrable discrete equations to be constructed from any element of infinite order in the symmetry group and still have the full parameter freedom for their surface type. We then also make the first steps toward a geometric theory of delay-differential Painlevé equations by giving a description of singularity confinement in this setting in terms of mappings between jet spaces

Special functions arising from discrete Painlevé equations: A survey

AbstractThis article is a survey on recent studies on special solutions of the discrete Painlevé equations, especially on hypergeometric solutions of the q-Painlevé equations. The main part of this survey is based on the joint work [K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Hypergeometric solutions to the q-Painlevé equations, IMRN 2004 47 (2004) 2497–2521, K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Construction of hypergeometric solutions to the q-Painlevé equations, IMRN 2005 24 (2005) 1439–1463] with Kajiwara, Masuda, Ohta and Yamada. After recalling some basic facts concerning Painlevé equations for comparison, we give an overview of the present status of studies on difference (discrete) Painlevé equations as a source of special functions