Holomorphic Motions and Structural Stability for Polynomial Automorphisms of C

Abstract

Combining ideas from real dynamics on compact manifolds and complex dynamics in one variable, we prove the structural stability of hyperbolic polynomial automorphisms in C. We consider families of hyperbolic polynomial automorphisms depending holomorphically on the parameter λ. This is done over a series of steps given a family {fλ}, where |λ| is sufficiently small, we construct mappings defined on a neighborhood U of J0 which conjugate f0 and fλ. Moreover, it is shown that J moves holomorphically. This conjugacy is then used to construct a conjugacy between f0 and fλ defined on a neighborhood M of J 0 ∪J− 0 . Finally, we extend such a mapping to construct a conjugacy on all of C.

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Cite this paper

@inproceedings{BuzzardHolomorphicMA, title={Holomorphic Motions and Structural Stability for Polynomial Automorphisms of C}, author={Gregery T. Buzzard and Adrian Jenkins} }