For any discrete time dynamical system with a rational evolution, we define an entropy, which is a global index of complexity for the evolution map. We analyze its basic properties and its relations to the singularities and the irreversibility of the map. We indicate how it can be exactly calculated. Postal address: Laboratoire de Physique Théorique et des Hautes EnergiesUniversité Pierre et Marie Curie, boîte postale 126.

We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are nonintegrable. As a more sensitive integrability test, we propose the analysis of the complexity ("algebraic entropy") of the map using the growth of the degree of its iterates: integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems.