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Axiom A polynomial skew products of ℂ2 and their postcritical sets

Published online by Cambridge University Press:  15 September 2008

LAURA DEMARCO  and
SUZANNE LYNCH HRUSKA
LAURA DEMARCO
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA (email: [email protected])
SUZANNE LYNCH HRUSKA
Affiliation:
Department of Mathematical Sciences, University of Wisconsin Milwaukee, PO Box 413, Milwaukee, WI 53201, USA (email: [email protected])
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1749 - 1779
Copyright
Copyright © 2008 Cambridge University Press

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