COMPLEX HORSESHOES AND THE DYNAMICS OF MAPPINGS OF TWO COMPLEX VARIABLES |
Posted on:1988-10-10 | Degree:Ph.D | Type:Dissertation |
University:Cornell University | Candidate:OBERSTE-VORTH, RALPH WERNER | Full Text:PDF |
GTID:1470390017957256 | Subject:Mathematics |
Abstract/Summary: | |
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 Henon mappings in two variables as well as higher dimensional analogues. From polynomial-like mappings, Henon-like and quasi-Henon-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. |
Keywords/Search Tags: | Horseshoes, Mappings, Complex, Dynamics |
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