Syst\`emes aux q-diff\'erences singuliers r\'eguliers: solutions canoniques, classification, matrice de connexion et monodromie

Abstract

G.D. Birkhoff extended the classical Riemann-Hilbert problem for differential equations to the case of ``fuchsian'' linear $q$-difference systems with rational coefficients. He solved it in the generic case: the classifying object which he introduces is made up of the connection matrix $P$, together with the exponents at 0 and $\infty$. We follow his method in the general case, but treat symetrically 0 and $\infty$ and use no ``wildly'' growing solutions. When $q$ tends to 1, $P$ tends to a locally constant matrix $\tilde{P}$ such that the (finitely many) values $\tilde{P}(a)^{-1}\tilde{P}(b)$ are the monodromy matrices of the limiting differential system (assumed to be non resonant at 0 and $\infty$) at the singularities on $\mathbf{C}^{*}$. This text is that of preprint 148 of the Laboratoire Emile Picard (february 1999). A shorter version was published by the Annales de l'Institut Fourier, 50, 4, (2000).