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 leafwise intersection points. Moreover, any Stein filling of dimension at least six has infinite-dimensional symplectic homology.
"Non-displaceable contact embeddings and infinitely many leaf-wise intersections." J. Symplectic Geom. 9 (3) 271 - 284, September 2011.