  Investigation of the effects of a contact surgery construction and of
invariance of contact homology reveals a rich new field of inquiry at the
intersection of dynamical systems and contact geometry. We produce contact
3-flows not topologically orbit-equivalent to any algebraic flow, including
examples on many hyperbolic 3-manifolds, and we show how the surgery produces
dynamical complexity for any Reeb flow compatible with the resulting contact
structure. This includes exponential complexity when neither the surg-ered flow
nor the surgered manifold are hyperbolic. We also demonstrate the use in
dynamics of contact homology, a powerful tool in contact geometry.
