Axiom A polynomial skew products of 2 and their postcritical sets

Laura Demarco*, Suzanne Lynch Hruska

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1749-1779
Number of pages31
JournalErgodic Theory and Dynamical Systems
Volume28
Issue number6
DOIs
StatePublished - Jan 1 2008

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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