Trivial dynamics for a class of analytic homeomorphisms of the plane

Rafael Ortega

doi:10.3934/dcdsb.2008.10.651

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2008, Volume 10, Issue 2&3: 651-659. Doi: 10.3934/dcdsb.2008.10.651

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Trivial dynamics for a class of analytic homeomorphisms of the plane

Rafael Ortega^1,

1.
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada

Received: August 31, 2006

Revised: June 30, 2007

Published: August 2008

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Abstract

A homeomorphism of the plane $h$ has trivial dynamics if every positive orbit $\{ h^n (p)\}_{n\geq 0}$ is either convergent (to a fixed point) or divergent (to infinity). The main result of this paper says that the property of trivial dynamics can be decided by computing the topological degree of $id -h$. In this result it is assumed that $h$ is analytic in the real sense. Some applications to difference equations and to periodic Newtonian differential equations are obtained.

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Mathematics Subject Classification: Primary: 37E30; Secondary: 39A11, 34C15.

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