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

In the current paper we study the q-analogue introduced by Jimbo and Sakai of the well known Painlevé 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 VI admits the differential Okamoto space via some natural limit process.