Abstract
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.
| Original language | American English |
|---|---|
| Pages (from-to) | 271-317 |
| Number of pages | 47 |
| Journal | Journal of the European Mathematical Society |
| Volume | 21 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2019 |
Bibliographical note
Funding Information: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.
Publisher Copyright:
© European Mathematical Society 2019.
Keywords
- Borel dynamics
- Entropy
- Ergodic theory
- Generators