In general, it is a difficult problem to understand how the Floer theory of a pair $(H, J)$ relates to the qualitative dynamics of the isotopy generated by $H$. In this talk I will discuss the question of interpreting (Hamiltonian) Floer-theoretic objects in terms of topological dynamics in low dimensions, together with the relationship of this question to a certain picture of how Hamiltonian Floer theory may be combined with the theory of finite energy foliations introduced by Hofer-Wysocki-Zehnder in order to gain an understanding of the structure of Hamiltonian systems on surfaces (via a Floer-theoretic construction of certain foliations introduced by Le Calvez in his study of surface homeomorphisms).