Ergodic Theory and Dynamical Systems

Research Article

Axiom A polynomial skew products of xs21022 and their postcritical sets

LAURA DEMARCOa1 and SUZANNE LYNCH HRUSKAa2

a1 Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA (email: demarco@math.uic.edu)

a2 Department of Mathematical Sciences, University of Wisconsin Milwaukee, PO Box 413, Milwaukee, WI 53201, USA (email: shruska@uwm.edu)

Abstract

A polynomial skew product of xs21022 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 xs21192 of degree at least two. For polynomial maps of xs2102, 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.

(Received April 23 2007)

(Revised October 19 2007)