Hofer growth of $C^1$-generic Hamiltonian flows

Abstract

We prove that on certain closed symplectic manifolds a C 1 -generic cyclic subgroup of the universal cover of the group of Hamiltonian diffeomorphisms is undistorted with respect to the Hofer metric.

Symplectomorphisms with positive metric entropy

“…Note that there are obstructions to partial hyperbolicity on certain symplectic manifolds (see [13] for a discussion); for example, surfaces different from $$\mathbb {T}^2$$ do not carry an Anosov diffeomorphism, and $${\mathbb {C}}{\mathbb {P}}^n$$ does not carry a partially hyperbolic symplectomorphism. For these manifolds, Theorem A implies that the $$C^1$$‐generic symplectomorphism has volume entropy 0.…”