From the Theorem of Ważewski to Computer Assisted Proofs in Dynamics

1. Introduction. In 1947 Tadeusz Wa˙ zewski [35, 36] proved his famous Retract Theorem. Roughly speaking the theorem states that if the trajectories of a flow leave a closed set W through a subset W − of the boundary of W , do it " transversally " , and the set W − is not a deformation retract of W then there exists a trajectory of the flow entirely contained in W. (We put the word trans-versally in quotation marks, because what really matters is a certain topological type of transversality.) The theorem soon found many applications in differential equations concerning the existence of bounded trajectories.