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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
Affiliation:
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)
SUZANNE LYNCH HRUSKA
Affiliation:
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 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
Copyright
Copyright © 2008 Cambridge University Press

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References

[1] Carleson, L. and Gamelin, T. W.. Complex Dynamics (Universitext: Tracts in Mathematics). Springer, New York, 1993.CrossRefGoogle Scholar
[2] Comerford, M.. Hyperbolic non-autonomous Julia sets. Ergod. Th. & Dynam. Sys. 26(2) (2006), 353377.CrossRefGoogle Scholar
[3] Diller, J. and Jonsson, M.. Topological entropy on saddle sets in P2. Duke Math. J. 103(2) (2000), 261278.CrossRefGoogle Scholar
[4] Fornæss, J. E. and Sibony, N.. Random iterations of rational functions. Ergod. Th. & Dynam. Sys. 11(4) (1991), 687708.CrossRefGoogle Scholar
[5] Fornæss, J. E. and Sibony, N.. Hyperbolic maps on P2. Math. Ann. 311(2) (1998), 305333.CrossRefGoogle Scholar
[6] Fornæss, J. E. and Sibony, N.. Dynamics of P2 (examples). Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998) (Contemporary Mathematics, 269). American Mathematical Society, Providence, RI, 2001, pp. 4785.CrossRefGoogle Scholar
[7] Hruska, S. L.. Rigorous numerical studies of the dynamics of polynomial skew products of C2. Complex Dynamics (Proceedings of an AMS–IMS–SIAM Joint Summer Research Conference on Complex Dynamics, 13 – 17 June, 2004) (Contemporary Mathematics, 396). Eds. R. Devaney and L. Keen. American Mathematical Society, Providence, RI, 2006, p. 208.Google Scholar
[8] Jonsson, M.. Holomorphic motions of hyperbolic sets. Michigan Math. J. 45(2) (1998), 409415.CrossRefGoogle Scholar
[9] Jonsson, M.. Dynamics of polynomial skew products on C2. Math. Ann. 314(3) (1999), 403447.CrossRefGoogle Scholar
[10] McMullen, C. T.. Complex Dynamics and Renormalization (Annals of Mathematics Studies, 135). Princeton University Press, Princeton, NJ, 1994.Google Scholar
[11] Nekrashevych, V.. An uncountable family of three-generated groups acting on the binary tree. Preprint, 2005.Google Scholar
[12] Papadantonakis, K. and Hubbard, J. H.. The FractalAsm page.http://www.math.cornell.edu/∼dynamics/FA.Google Scholar
[13] Robinson, C.. Dynamical Systems, 2nd edn (Stability, Symbolic Dynamics, and Chaos). CRC Press, Boca Raton, FL, 1999.Google Scholar
[14] Sullivan, D. P. and Thurston, W. P.. Extending holomorphic motions. Acta Math. 157(3–4) (1986), 243257.CrossRefGoogle Scholar
[15] Sumi, H.. Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. & Dynam. Sys. 21(2) (2001), 563603.CrossRefGoogle Scholar
[16] Sumi, H.. Semi-hyperbolic fibered rational maps and rational semigroups. Ergod. Th. & Dynam. Sys. 26(3) (2006), 893922.CrossRefGoogle Scholar
[17] Sumi, H.. Dynamics of postcritically bounded polynomial semigroups. Preprint, arXiv:math.DS/0703591, submitted, February 15, 2007.Google Scholar