NON-DISPLACEABLE CONTACT EMBEDDINGS AND INFINITELY MANY LEAF-WISE INTERSECTIONS

We construct using Lefschetz fibrations a large family of contact manifolds with the following properties: any bounding contact embedding into an exact symplectic manifold satisfying a mild topological assumption is non-displaceable and generically has infinitely many leaf-wise intersection points. Moreover, any Stein filling of dimension at least six has infinite-dimensional symplectic homology.