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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $f \\colon X \\to X$ be a [[continuous map|0VGFHTM75Y]].\nThe map $f$ is *topologically mixing* if for all nonempty [[open sets|02BH3MCBQB]] $U, V \\subseteq X$ there is $N \\ge 1$ with $f^n(U) \\cap V \\neq \\emptyset$ for every $n \\ge N$.\n\n## Remarks\n\nEvery topologically mixing map is [[topologically transitive|FR49Y9GQHR]].\nExample: the doubling map $f(x) = 2x \\bmod 1$ on the circle $\\R / \\Z$.\nA nonempty open set $U$ contains an open arc of length $\\ell \u003e 0$; choose $k$ with $2^k \\ell \u003e 1$.\nThe map $f^n$ multiplies lengths of small arcs by $2^n$, so for $n \\ge k$ the image $f^n(U)$ is the whole circle and meets every $V$.\nNon-example: the rotation $f(x) = x + \\alpha \\bmod 1$ with $\\alpha$ irrational, and $U = V = (-1/8, 1/8)$.\nHere $f^n(U) \\cap U \\neq \\emptyset$ only when $n\\alpha$ is within $1/4$ of an integer.\nBy Weyl's equidistribution theorem, about half of all $n$ fail this, so there is no $N$ as required.\n\n## Explanations\n\n### L1 human\n\nA mixing map brings each open region into each other open region and keeps it meeting that region at all large times.\n",
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