  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.
