Abstract. 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 [19].
We study certain topological entropy-type growth characteristics of Hamiltonian flows of a special form on the cotangent bundle T * M over a closed Riemannian manifold (M, g). More precisely, the Hamiltonians generating the flows have to be of the form H(t, q, p) = f (|p|) for |p| ≥ 1, and the growth characteristics reflect how the Riemannian volumes of the unit balls in the fibers of T * M grow with the flow. Using a version of Lagrangian Floer homology and recent work of Abbondandolo and Schwarz, we obtain uniform lower bounds for the growth characteristics that depend only on (M, g) or, more precisely, on the homology of the sublevel sets of the energy functional defined by the Riemannian metric g on the space of based loops of Sobolev class W 1,2 in M . These lower bounds, in turn, can be estimated from below by purely topological characteristics of M . Our results in particular refine previous results of Dinaburg, Gromov, Paternain, and Paternain-Petean on the topological entropy of geodesic flows.As an application, we obtain that for Riemannian manifolds, all of whose geodesics are closed (so-called P -manifolds), the fiberwise volume growth of every symplectomorphism in the symplectic isotopy class of the Dehn-Seidel twist is at least linear. This extends the main result of our paper [30] from the class of all currently known P -manifolds to all P -manifolds.
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