Slow entropy and symplectomorphisms of cotangent bundles

We consider an entropy-type invariant which measures the polynomial volume growth of submanifolds under the iterates of a map, and we establish sharp uniform lower bounds of this invariant for the following classes of symplectomorphisms of cotangent bundles over a compact base:

1. non-identical compactly supported symplectomorphisms which are symplectically isotopic to the identity,
2. symplectomorphisms generated by classical Hamiltonian functions,
3. Dehn twist like symplectomorphisms over compact rank one symmetric spaces.