Fiberwise volume growth via Lagrangian intersections

We consider Hamiltonian diffeomorphisms $\phi$ of the unit cotangent bundle over a closed Riemannian manifold $(M,g)$ which extend to Hamiltonian diffeomorphisms of $T^*M$ equal to the time-1-map of the geodesic flow for $|p| \ge 1$. For such diffeomorphisms we establish uniform lower bounds for the fiberwise volume growth of $\phi$ which were previously known for geodesic flows and which depend only on $(M,g)$ or on the homotopy type of $M$. More precisely, we show that for each $q \in M$ the volume growth of the unit ball in $T_q^*M$ under the iterates of $\phi$ is at least linear if $M$ is rationally elliptic, is exponential if $M$ is rationally hyperbolic, and is bounded from below by the growth of the fundamental group of $M$. In the case that all geodesics of $g$ are closed, we conclude that the slow volume growth of every symplectomorphism in the symplectic isotopy class of the Dehn--Seidel twist is at least 1, completing the main result of \cite{FS:GAFA}. The proofs use the Lagrangian Floer homology of $T^*M$ and the Abbondandolo--Schwarz isomorphism from this homology to the homology of the based loop space of $M$.