Qualitative Theory of Dynamical Systems

, Volume 10, Issue 2, pp 247–275 | Cite as

Real Dynamics of Integrable Birational Maps

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Abstract

We begin by studying the dynamic generated by iteration of birational maps in \({{\mathbb R}^k}\) with k − 1 independent rational first integrals. We prove that each level curve can be desingularized and compactified being homeomorphic to a finite union of disjoint circles and open intervals. Furthermore, the map can be extended homeomorphically in a natural way to this space. After, we focus our attention in the case that the map has a rational invariant measure and we see that in most cases the orbit of a point or it is periodic or it fulfills densely some connected components of its corresponding level set. Some applications in dimension two and three are presented.

Keywords

Birational maps Circle maps Difference equations Discrete dynamical systems First integrals Integrable maps Lie-symmetries Periodic orbits Rotation numbers 

Mathematics Subject Classification (2000)

39A11 39A20 
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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Dept. de Matemàtiques, Facultat de CiènciesUniversitat Autònoma de BarcelonaBarcelonaSpain

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