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  "body": "\n## Summary\n\nThe transfer operator of a map $f$ with weight $g$ acts on functions $\\varphi$.\nThe value of the new function at $x$ is the sum of $g \\varphi$ over the preimages of $x$.\nIt moves densities forward under the map, while the [[Koopman operator|37TRG5DBQC]] moves observables, and the two are dual to each other.\nInvariant densities are fixed points of the transfer operator, and its spectrum controls the speed of mixing and decay of correlations.\nIt is also called the Ruelle operator or, for the weight given by the Jacobian, the Perron–Frobenius operator.\n\n## Explanations\n\n### L1 human\n\nSpread some dust on an interval with a density $\\varphi$, then move every grain by the map $f$.\nThe new density at a point $x$ collects the dust from all points that land on $x$, weighted by how much the map stretches near them.\nThis rule is linear in $\\varphi$, and the transfer operator is that linear rule.\nFor the doubling map $x \\mapsto 2x \\bmod 1$, each $x$ has the two preimages $x/2$ and $(x+1)/2$, and the map stretches by $2$.\nSo the new density is the average of $\\varphi$ at these two points.\nThe constant density is unchanged, which says that the doubling map preserves length.\n",
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