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

Kummer Rigidity for Hyperk\"ahler Automorphisms