Invariance of the Gibbs Measure for the Schrödinger-Benjamin-Ono System

We prove the invariance of the Gibbs measure for the periodic SchrödingerBenjamin-Ono system (when the coupling parameter |γ| 6= 0, 1) by establishing a new local well-posedness in a modified Sobolev space and constructing the Gibbs measure (which is in the sub-L setting for the Benjamin-Ono part.) We also show the ill-posedness result in H(T)×H 1 2 (T) for s < 1 2 when |γ| 6= 0, 1 and for any s ∈ R when |γ| = 1.