In the late 90's Hofer-Wysocki-Zehnder showed that certain 3-dimensional Hamiltonian energy surfaces admit a surprising geometric decomposition by surfaces transverse to the dynamics. Their approach used holomorphic curve methods from symplectic geometry. Independently in 2004 LeCalvez constructed objects of a similar flavour for 2-dimensional systems using entirely topological methods (in particular without pde's) and successfully applied these to the dynamics of surface homeomorphisms. In this talk I hope to briefly discuss 1) the two points of view, 2) how a notion from LeCalvez work suggests interesting new possibilities in global symplectic tools in low dimensions such as Floer theory (work in progress), 3) some applications to surface dynamics.

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