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  "body": "\n## Summary\n\nThe Kolmogorov–Sinai entropy, or metric entropy, of a [[measure-preserving transformation|556WBDAHNP]] measures how much new information each step of the map produces.\nIt is defined from the Shannon entropy of finite partitions, refined along the orbit and taken per step.\nIt is an isomorphism invariant.\nIt tells apart systems with the same spectral data, such as two Bernoulli shifts whose probability vectors have different entropies.\nBy the [[Ornstein isomorphism theorem|8H4GF58E9M]] it is a complete invariant for Bernoulli shifts.\n\n## Explanations\n\n### L1 human\n\nWatch a system through a coarse lens that only tells you in which of a few boxes the point is.\nRecord the sequence of boxes visited over many steps.\nThe entropy is the number of bits per step that you need, in the long run, to write down this record, for the best choice of boxes.\nTossing a fair coin produces $\\log 2$ of entropy per step, one bit.\nA rotation of a circle produces none, because after a while the past record predicts the future.\n",
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