Kummer Rigidity for Hyperk\"ahler Automorphisms

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.