Complex Horseshoes and the Dynamics of Mappings of Two Complex Variables

Abstract

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.