We show that if an automorphism of a standard Borel space does not admit finite invariant measures, then it has a two-set generator. This implies that if the entropies of invariant probability measures of a Borel system are all less than log k, then the system admits a k-set generator, and that a wide class of hyperbolic-like systems are classified completely at the Borel level by entropy and periodic points counts.
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Acknowledgments. I would like to thank A. Kechris and the anonymous referee for pointing out that there is no need to exclude wandering sets in Theorem 1.1. I am also grateful to the referee for a very perceptive and careful reading of the paper, which has led to a much improved manuscript. This research was partially supported by ISF grant 1409/11 and ERC grant 306494.
© European Mathematical Society 2019.
- Borel dynamics
- Ergodic theory