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  "body": "\n## Example\n\nThe *dyadic transformation*, or *doubling map*, is\n$$T \\colon [0, 1) \\to [0, 1), \\qquad T(x) = 2x \\bmod 1.$$\n\nIt is a [[measure-preserving transformation|556WBDAHNP]] for [[Lebesgue measure|2Z2QVWZ262]] $\\lambda$.\nCheck on an interval $[a, b) \\subseteq [0, 1)$:\n$$T^{-1}[a, b) = \\Big[\\frac{a}{2}, \\frac{b}{2}\\Big) \\cup \\Big[\\frac{a+1}{2}, \\frac{b+1}{2}\\Big),$$\na disjoint union of two intervals of length $(b - a)/2$, so $\\lambda(T^{-1}[a, b)) = b - a$.\n\nWrite $x = \\sum_{n \\ge 1} x_n 2^{-n}$ in binary, with digits $x_n \\in \\set{0, 1}$ not eventually all equal to $1$.\nThen $T$ deletes the first digit: $T(x) = \\sum_{n \\ge 1} x_{n+1} 2^{-n}$.\nUnder $\\lambda$ the digits are independent fair coin tosses.\nSo the binary expansion is an isomorphism, up to null sets, from $T$ to the one-sided [[Bernoulli scheme|D33JK3AXRZ]] $B(1/2, 1/2)$.\nHence $T$ is [[mixing|S3RJQ9CX2P]], and its [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] is $\\log 2$.\n\nThe map is not invertible: each point has two preimages.\n\n## Explanations\n\n### L1 human\n\nDouble the number and throw away the whole part.\nIn binary, this is the same as shifting the digits one step to the left and forgetting the first digit.\nSo the future of a point is written in its digits, and for a typical number these digits look like fair coin tosses.\nThat is why such a simple formula gives a fully random system.\n",
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