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  "body": "\n## Definition\n\nLet $X$ be a set, let $T \\colon X \\to X$ be a map with [[iterates|AYVTR2ES5B]] $T^k$, and let $f \\colon X \\to \\R$ be a [[function|0EPAT51DXQ]].\nFor an integer $n \\ge 1$, the *Birkhoff sum* of $f$ is the function\n$$S_n f(x) = \\sum_{k=0}^{n-1} f(T^k(x)), \\qquad x \\in X,$$\nand the *Birkhoff average*, or ergodic average, of $f$ is\n$$A_n f(x) = \\frac{1}{n} S_n f(x).$$\n\n## Remarks\n\nThe same formulas make sense for $f$ with values in $\\C$ or in any real vector space.\nThe sums satisfy the cocycle relation $S_{m+n} f = S_m f + (S_n f) \\circ T^m$ for all $m, n \\ge 1$.\nIf $\\mu$ is a $T$-invariant probability measure and $f$ is integrable, then $\\int A_n f \\, d\\mu = \\int f \\, d\\mu$ for every $n$.\n\nExample: let $T(x) = x + 1/2 \\bmod 1$ on $[0, 1)$ and $f(x) = x$.\nFor $x \\in [0, 1/2)$ the orbit alternates between $x$ and $x + 1/2$, so $S_{2m} f(x) = m(2x + 1/2)$ and $A_{2m} f(x) = x + 1/4$.\n\nNon-example: the space average $\\int_0^1 f(x) \\, dx = 1/2$ is not a Birkhoff average, since it does not depend on a starting point; in this example $A_{2m} f(x) = x + 1/4$ differs from $1/2$ unless $x = 1/4$.\n",
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