Fredholm theory and transversality for noncompact pseudoholomorphic curves in symplectizations

We study pseudoholomorphic maps from a punctured Riemann surface into the symplectization of a contact manifold. A Fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the Euler characteristic of the Riemann surface and the asymptotic data given by the periodic solutions of the Reeb vector field associated to the contact form. The transversality results, establish the existence of additional structure for these spaces. To be more precise we prove that these spaces are generically smooth manifolds and therefore their virtual dimension coincides with their actual dimension.