In this paper, we explore loops of non-autonomous Hamiltonian diffeomorphisms with degenerate fixed maxima. We show that such loops can not have totally degenerate fixed global maxima. This has applications for the Hofer geometry of the group of Hamiltonians for certain symplectic four manifolds and also gives criteria for certain four manifolds to be uniruled.
"Degenerate maxima in Hamiltonian systems." J. Symplectic Geom. 10 (1) 1 - 16, March 2012.