Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation

Abstract: In the current paper we study the $q$-analogue introduced by Jimbo and Sakai of the well known Painlev\'e VI differential equation. We explain how it can be deduced from a $q$-analogue of Schlesinger equations and show that for a convenient change of variables and auxiliary parameters, it admits a $q$-analogue of Hamiltonian formulation. This allows us to show that Sakai's $q$-analogue of Okamoto space of initial conditions for $qP_\mathrm{VI}$ admits the differential Okamoto space \emph{via} some natural limit process.