SIMPLE AND COMPLEX DYNAMICS FOR CIRCLE MAPS* LLUÍS ALSEDÁ AND VLADIMIR FEDORENKO The continuous self maps of a closed interval of the real line with zero topological entropy can be characterized in terms of the dynamics of the map on its chain recurrent set. In this paper we extend this characterization to continuous self maps of the circle . We show that, for these maps, the chain recurrent set can exhibit a new dynamic behaviour which is specific of the circle maps of degree one.