Morse decomposition separates gradient dynamics from recurrent dynamics by gathering all the recurrent behavior in isolated invariant sets known as Morse sets. However, the nature of recurrence inside Morse sets remains hidden. A more general concept of Morse predecomposition aims to reveal not only the separation but also the internal structure of Morse sets. I'll present it in the setting of combinatorial dynamical systems where the concept is easier to grasp, as well as in the setting of classical dynamics. I'll also present a potential of Morse predecompositions in combinatorial dynamics in providing new methods for computer assisted proofs of the existence of periodic orbits and chaotic invariant sets in classical dynamical systems.

Remark: This seminar is a joint seminar with Kyoto University Applied Mathematics Seminar.