Axiom A polynomial skew products of C^2 and their postcritical sets

Details Axiom A polynomial skew products of C^2 and their postcritical sets Axiom A polynomial skew products of C^2 and their postcritical sets

2007-04-06

arxiv.org/abs/0704.0942v2

Ergodic Theory and Dynamical Systems, Volume 28, Issue 06, Dec. 2008, pages 1749--1779.

Mathematics

Dynamical Systems

33 pages, 3 figures

Scientific paper

A polynomial skew product of C^2 is a map of the form f(z,w) = (p(z), q(z,w)), where p and q are polynomials, such that f is regular of degree d >= 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 (Math. Ann., 1999) 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 the 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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