Dynamics of Polynomial Diffeomorphisms

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Some of the most impressive recent work in dynamical systems has been an outgrowth of the study of dynamics in one complex variable. The fundamental assumption motivating this proposal is that complex methods have an important role to play in dynamics when the number of dimensions is greater than one as well. The subject of this proposal is a particular "model family" of dynamical systems, the polynomial diffeomorphisms in two complex dimensions. The complex Henon diffeomorphisms which are a notable special case. These are perhaps the simplest invertible holomorphic dynamical system with interesting dynamics. One of the lessons of dynamics in one variable is that there is a range of dynamical behaviors starting with expanding maps and continuing with Misurewicz maps, semi-hyperbolic and Collet-Eckmann maps. The two variable analog of the expanding property is hyperbolicity. This proposal is focused on understanding the two variable analog of the semi-hyperbolic condition which we call quasi-hyperbolicity. In one variable these conditions are related to the recurrence properties of critical points. In two variables the notion of critical point needs to be replaced by other concepts such as regularity of stable and unstable manifolds. An interesting example of quasi-hyperbolic diffeomorphisms are real polynomial diffeomorphisms of maximal entropy such as limits of horseshoes in the real Henon case. Though several questions about the behavior of such diffeomorphisms have been answered in previous work a number of open questions remain.The introduction of the computer has increased the usefulness of simple deterministic mathematical models in a number of sciences. An illustrative example is the logistic map which can be used to describe the behavior of a single insect population in successive years. When the mathematical model is linear there is a well developed underlying theory. When the model is non-linear there are important theoretical questions which we have not yet been able to address. Recently mathematicians have made important breakthroughs in understanding the logistic map. One technique which proved essential was the consideration of an associated complex dynamical system. My proposal addresses some of the problems of using similar complex techniques when the system involved has dimension greater than one (for example when there are two interacting populations.)