Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system

We prove the invariance of the Gibbs measure for the periodic Schrodinger-Benjamin-Ono system (when the coupling parameter vertical bar gamma vertical bar not equal 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(2) setting for the Benjamin Ono part). We also show the ill-posedness result in H(s)(T) x H(s-1/2)(T) for s <1/2 when vertical bar gamma vertical bar not equal 0, 1 and for any s is an element of R when vertical bar gamma vertical bar = 1.