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. 
Urs Frauenfelder, Felix Schlenk