An ergodic system is dominant exactly when it has positive entropy

Abstract An ergodic dynamical system $\mathbf {X}$ is called dominant if it isomorphic to a generic extension of itself. It was shown by Glasner et al [On some classes measure preserving transformations. Trans. Moscow Math. Soc. 82 (1) (2021), 15–36] that Bernoulli systems with finite entropy are dominant. In this work, we show first every positive dominant, and then has zero entropy, not