  In this study, a theory analogous to both the theories of polynomial-like
mappings and Smale's real horseshoes is developed for the study of the dynamics
of mappings of two complex variables.
  In partial analogy with polynomials in a single variable there are the
H\'enon mappings in two variables as well as higher dimensional analogues. From
polynomial-like mappings, H\'enon-like and quasi-H\'enon-like mappings are
defined following this analogy. A special form of the latter is the complex
horseshoe.
  The major results about the real horseshoes of Smale remain true in the
complex setting. In particular:
  (1) Trapping fields of cones(which are sectors in the real case) in the
tangent spaces can be defined and used to find horseshoes.
  (2) The dynamics of a horseshoe is that of the two-sided shift on the symbol
space on some number of symbols which depends on the type of the horseshoe.
  (3) Transverse intersections of the stable and unstable manifolds of a
hyperbolic periodic point guarantee the existence of horseshoes.
