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

## TL;DR

This paper extends the understanding of polynomial skew products in C^2, showing that critical orbits either escape or accumulate on attracting sets, and introduces new examples of Axiom A maps with diverse postcritical behaviors.

## Contribution

It establishes an analogue of hyperbolicity criteria for polynomial skew products and constructs new Axiom A examples with varied postcritical dynamics.

## Key findings

- Critical orbits escape or accumulate on attracting sets.
- Axiom A maps characterized by postcritical set behavior.
- New examples of Axiom A maps with diverse postcritical behaviors.

## Abstract

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.