Axiom a Polynomial Skew Products of C2 and Their Postcritical Sets

A polynomial skew product of C 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 P of degree ≥ 2. 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 ([Jon99]) 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.