Abstract
We show that every finite entropy ergodic transformation can be represented as a Lebesgue measure-preserving homeomorphism of the twodimensional torus. Whether this is possible with diffeomorphisms is still unknown. Our proof suggests a notion of “universality” for homeomorphisms, about which we formulate several natural but unanswered questions.
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Both authors were supported in part by National Science Foundation Grant MCS 76-09159.
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Lind, D.A., Thouvenot, J.P. Measure-preserving homeomorphisms of the torus represent all finite entropy ergodic transformations. Math. Systems Theory 11, 275–282 (1977). https://doi.org/10.1007/BF01768481
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DOI: https://doi.org/10.1007/BF01768481