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  "body": "\n## Example\n\nFor a parameter $\\mu \\in [0, 2]$, the *tent map* is\n$$T_\\mu \\colon [0, 1] \\to [0, 1], \\qquad T_\\mu(x) = \\mu \\min(x, 1 - x).$$\nIts graph is a tent: it rises with slope $\\mu$ on $[0, 1/2]$ and falls with slope $-\\mu$ on $[1/2, 1]$.\nIts top value is $\\mu/2 \\le 1$, so it maps $[0, 1]$ into itself and is a [[discrete-time dynamical system|5AA9AJEARJ]].\n\nCheck that [[Lebesgue measure|2Z2QVWZ262]] $\\lambda$ is [[invariant|B232EZ1FMB]] for $\\mu = 2$.\nFor $0 \\le a \u003c b \\le 1$, the preimage $T_2^{-1}([a, b])$ is $[a/2, b/2] \\cup [1 - b/2, 1 - a/2]$, two intervals of length $(b - a)/2$ each.\nSo $\\lambda(T_2^{-1}[a, b]) = b - a = \\lambda([a, b])$, and this extends from intervals to all Borel sets.\n\n## Remarks\n\nFor $\\mu = 2$ the map is ergodic for Lebesgue measure, has [[topological entropy|BDZYBXBFDE]] $\\log 2$, and has [[Lyapunov exponent|NJ1FJSNK6A]] $\\log 2$ at every point whose orbit avoids $1/2$, since $\\abs{T_2'} = 2$.\nFor $0 \\le \\mu \u003c 1$ every orbit tends to the fixed point $0$.\nThe map $T_2$ is conjugate to the logistic map $x \\mapsto 4x(1 - x)$.\n\n## Explanations\n\n### L1 human\n\nThe tent map stretches the unit interval to twice its length and folds it back onto itself, like kneading dough.\nEach step doubles small distances between nearby points, so tiny errors grow fast and long-term prediction fails.\nYet the stretching and folding is so even that a random starting point spends equal time in intervals of equal length.\n",
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