Dynamics of Polynomial Diffeomorphisms

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Abstract: This project involves developing analogs of some of the important concepts in the theory of one dimensional complex dynamics in the context of diffeomorphisms in two complex dimensions. One powerful tool in dynamics is hyperbolicity. In one complex dimension this property is easily characterized in terms of the behavior of the critical point. The first part of this project involves characterizations of hyperbolicity and related properties for diffeomorphisms in two complex variables with the expectation that this work will shed light both on the hyperbolic and the non-hyperbolic cases. The proposed characterizations involve the regularity of unstable manifolds, the geometry of the intersection of unstable manifolds with the set of bounded orbits and growth conditions on Green functions in unstable manifolds. A second set of questions involve symbolic dynamics. There are situations in which external rays can be used to parametrize ``Julia sets''. This gives a new way of symbolically describing Julia sets in many cases. We raise questions about which symbolic descriptions can occur. A third set of questions involve the property of ``connectivity of the Julia set'' and the properties of the connectivity locus. Dynamical systems is an area of pure mathematics which deals with theoretical questions that arise in studying long term behavior of mathematical models of physical systems. We restrict ourselves to models whose state at any time can be described by a finite number of variables. Even though these systems can display complicated chaotic behavior, there are some powerful theoretical tools that can be applied to study them in some cases. On the other hand there are many cases where these tools do not apply. Recently new techniques have been developed which use complex analysis but these techniques have so far been limited to one variable whereas typical dynamical systems involve many variables. The proposed research aims to establish a link between dynamics of one complex v ariable and the more typical dynamical systems which involve many variables.