  We introduce the notion of asymptotically finitely generated contact
structures, which states essentially that the Symplectic Homology in a certain
degree of any filling of such contact manifolds is uniformly generated by only
finitely many Reeb orbits. This property is used to generalize a famous result
by Ustilovsky: We show that in a large class of manifolds (including all unit
cotangent bundles and all Weinstein fillable contact manifolds with torsion
first Chern class) each carries infinitely many exactly fillable contact
structures. These are all different from the ones constructed recently by
Lazarev. Along the way, the construction of Symplectic Homology is made more
general. Moreover, we give a detailed exposition of Cieliebak's Invariance
Theorem for subcritical handle attaching, where we provide explicit
Hamiltonians for the squeezing on the handle.
