A polynomial skew product of ℂ2 is a map of the form f(z,w)=(p(z),q(z,w)), where p and q are polynomials, such that f extends holomorphically to an endomorphism of ℙ2 of degree at least two. For polynomial maps of ℂ, hyperbolicity is equivalent to the condition that the closure of the postcritical set is disjoint from the Julia set; further, critical points either iterate to an attracting cycle or infinity. For polynomial skew products, Jonsson [Dynamics of polynomial skew products on C2. Math. Ann. 314(3) (1999), 403–447] established that f is Axiom A if and only if the closure of the postcritical set is disjoint from the right analog of the Julia set. Here we present an analogous conclusion: critical orbits either escape to infinity or accumulate on an attracting set. In addition, we construct new examples of Axiom A maps demonstrating various postcritical behaviors.

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Abstract. For complex parameters a, c, we consider the Hénon mapping Ha,c : C 2 → C 2 , given by (x, y) → (x 2 + c − ay, x), and its Julia set, J. In this paper, we describe a rigorous computer program for attempting to construct a cone field in the tangent bundle over J, which is preserved by DH, and a continuous norm in which DH (and DH −1 ) uniformly expands the cones (and their complements). We show a consequence of a successful construction is a proof that H is hyperbolic on J. We give several new examples of hyperbolic maps, produced with our computer program, Hypatia, which implements our methods.

Abstract. We describe a rigorous computer algorithm for attempting to construct an explicit, discretized metric for which a polynomial map f : C → C is expansive on a neighborhood of the Julia set, J. We show construction of such a metric proves the map is hyperbolic. We also examine the question of whether the algorithm can be improved, and the related question of how to build a metric close to euclidean. Finally, we give several examples generated with our implementation of this algorithm.

Little is known about the global topology of the Fatou set U (f) for holomorphic endomorphisms f : CP k → CP k , when k > 1. Classical theory describes U (f) as the complement in CP k of the support of a dynamically defined closed positive (1, 1) current. Given any closed positive (1, 1) current S on CP k , we give a definition of linking number between closed loops in CP k \ supp S and the current S. It has the property that if lk(γ, S) = 0, then γ represents a non-trivial homology element in H1(CP k \ supp S). As an application, we use these linking numbers to establish that many classes of endomorphisms of CP 2 have Fatou components with infinitely generated first homology. For example, we prove that the Fatou set has infinitely generated first homology for any polynomial endomorphism of CP 2 for which the restriction to the line at infinity is hyperbolic and has disconnected Julia set. In addition we show that a polynomial skew product of CP 2 has Fatou set with infinitely generated first homology if some vertical Julia set is disconnected. We then conclude with a section of concrete examples and questions for further study.