  We show that a holomorphic automorphism on a projective hyperk\"ahler
manifold that has positive topological entropy and has volume measure as the
measure of maximal entropy, is necessarily a Kummer example, partially
extending the analogous results in (Cantat-Dupont 2020)(Filip-Tosatti 2018) for
complex surfaces.
  A trick with Jensen's inequality is used to show that stable and unstable
distributions exhibit uniform rate of contraction and expansion, and with them
our hyperk\"ahler manifold is shown to be flat. A result in
(Greb-Kebekus-Peternell 2016) then implies that our hyperk\"ahler manifold is
birational to a torus quotient, giving the Kummer example structure.
