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An ergodic system is dominant exactly when it has positive entropy
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 10 / October 2023
- Published online by Cambridge University Press:
- 04 November 2022, pp. 3216-3230
- Print publication:
- October 2023
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- Article
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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.
