On the Quadratic Endomorphisms of the Plane

In this article we establish the following result: if a nondegenerate quadratic endomorphism of the plane has no fixed points, then every point has empty omega-limit set and alpha-limit set. It is also shown that there exists a six parameter family open and dense in the space of all quadratic mappin...

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Bibliographic Details
Published in:Advanced nonlinear studies 2004-02, Vol.4 (1), p.37-55
Main Authors: Bofill, F., Garrido, J.L., Vilamajó, F., Romero, N., Rovella, A.
Format: Article
Language:English
Subjects:
Nondegenerate quadratic plane endomorphisms
α-limit set
ω-limit set
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Summary:In this article we establish the following result: if a nondegenerate quadratic endomorphism of the plane has no fixed points, then every point has empty omega-limit set and alpha-limit set. It is also shown that there exists a six parameter family open and dense in the space of all quadratic mappings of the plane (even those having fixed points). The degenerate case (when the quadratic forms of both components are linearly dependent), for which the theorem fails, is considered in the last section.
ISSN:1536-1365
2169-0375
DOI:10.1515/ans-2004-0103