Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation (Q6607738)

The \(q\)-analogue \(qP_{\mathrm{VI}}\) of the Painlevé VI equation introduced by Jimbo and Sakai is defined by means of the \(q\)-derivative of a function \(f\). This is the operator \(\partial_{q,t}:=(\sigma_{q,t}-1)/(q-1)t\), where \(\sigma_{q,t}\) associates to \(f(t)\) the function \(f(q\cdot t)\). The authors explain how \(qP_{\mathrm{VI}}\) can be deduced from a \(q\)-analogue of the Schlesinger equations (based on the Schlesinger \(q\)-isomonodromy) and show that for a convenient change of variables and auxiliary parameters, it admits a \(q\)-analogue of Hamiltonian formulation. This allows to show that Sakai's \(q\)-analogue of Okamoto space of initial conditions for \(qP_{\mathrm{VI}}\) admits the differential Okamoto space via some natural limit process.