Relative Hofer Geometry and the Asymptotic Hofer-Lipschitz Constant

Let $(M,omega)$ be a symplectic manifold and $U$ an open subset of  $M$. I
study the natural inclusion of the group of Hamiltonian  diffeomorphisms of
$U$ into the group of Hamiltonian diffeomorphisms  of $M$. The main result
is an upper bound for this map in terms of the  Hofer norms for $U$ and
$M$. Applications are upper bounds on the  relative Hofer diameter of $U$
and the asymptotic Hofer-Lipschitz  constant, which are often sharp up to
constant factors. As another  consequence, the relative Hofer diameter of
certain symplectic  submanifolds vanishes."