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We construct using Lefschetz fibrations a large family of contact manifoldswith the following properties: Any bounding contact embedding into an exactsymplectic manifold satisfying a mild topological assumption isnon-displaceable and generically has infinitely many leaf-wise intersectionpoints. Moreover, any Stein filling has infinite dimensional symplectichomology.

Non-displaceable contact embeddings and infinitely many leaf-wise intersections