  We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic
manifolds. To this end we first establish the Piunikhin-Salamon-Schwarz
isomorphism between the Floer homology and the Morse homology of such a
manifold, and then use this isomorphism to construct a biinvariant metric on
the group of compactly supported Hamiltonian diffeomorphisms analogous to the
metrics constructed by Viterbo and Schwarz. These tools are then applied to
prove and reprove results in Hamiltonian dynamics. Our applications comprise a
uniform lower estimate for the slow entropy of a compactly supported
Hamiltonian diffeomorphism, the existence of infinitely many nontrivial
periodic points of a compactly supported Hamiltonian diffeomorphism of a
subcritical Stein manifold, old and new cases of the Weinstein conjecture, and,
most noteworthy, new existence results for closed orbits of a charge in a
magnetic field on almost all small energy levels. We shall also obtain some old
and new Lagrangian intersection results. Applications to Hofer's geometry on
the group of compactly supported Hamiltonian diffeomorphisms will be given in a
sequel.
