Polynomial automorphisms of C^2 with the same equilibrium measure

We review the following question: when do two polynomial automorphisms of C^2 have the same maximal entropy measure? While the answer has been known for some time when both automorphisms are compositions of Hénon maps, the general case is more delicate. In this (mostly expository) note we explain how recent results in random dynamics give an almost complete solution to the question, and point out a number of related open problems.