On the Riemann-Hilbert Problem for a q-Difference Painlevé Equation

A Riemann-Hilbert problem for a q-difference Painlevé equation, known as $$q{\text {P}}_{{\text {IV}}}$$ , is shown to be solvable. This yields bijective correspondence between the transcendental solutions of and corresponding data on an associated q-monodromy surface. We also construct moduli space explicitly.