Real dynamics of integrable birational maps

We begin by studying the dynamic generated by iteration of birational maps in ℝk 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. © Springer Basel AG 2011.