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  1. ; ; ; (, Ergodic Theory and Dynamical Systems)
    An ergodic dynamical system $$\mathbf{X}$$ is called dominant if it is isomorphic to a generic extension of itself. It was shown by Glasner et al [On some generic classes of ergodic 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 that every ergodic system with positive entropy is dominant, and then that if $$\mathbf{X}$$ has zero entropy, then it is not dominant. 
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