Ornstein's Theorem states that any two ergodic measure-preserving transformations of a probability space are isomorphic if they have the same entropy. This fundamental result connects the concepts of ergodicity, measure theory, and entropy in dynamical systems, highlighting that systems with the same level of unpredictability (entropy) can be considered structurally the same. This theorem has opened up pathways for current research directions, particularly in understanding the classification of dynamical systems.
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