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  "body": "\n## Definition\n\nLet $X$ be a set and let $f \\colon X \\to X$ be a [[function|0EPAT51DXQ]].\nA *fractional iteration* of $f$ is a family $(f^t)_{t \\ge 0}$ of functions $f^t \\colon X \\to X$ such that:\n\n1. $f^0 = \\id_X$, the [[identity function|0PSKT5W817]] of $X$.\n2. $f^1 = f$.\n3. $f^s \\circ f^t = f^{s+t}$ for all real $s, t \\ge 0$.\n\n## Remarks\n\nExample: on $X = (0, \\infty)$, let $f(x) = \\frac{x}{1 + x}$ and $f^t(x) = \\frac{x}{1 + tx}$.\nClauses 1 and 2 hold by substitution, and clause 3 holds since $f^s(f^t(x)) = \\frac{x/(1+tx)}{1 + sx/(1+tx)} = \\frac{x}{1 + (s+t)x}$.\n\nNon-example: on $X = \\R$, the family $f^t(x) = (1 + t) x$ satisfies clauses 1 and 2 for $f(x) = 2x$, but fails clause 3, since $f^1(f^1(x)) = 4x \\ne 3x = f^2(x)$.\n\nClause 3 with integer $s, t$ forces $f^n$ to be the usual [[iterate|AYVTR2ES5B]], and $f^{1/n}$ is an [[iterative root|27C6J3YNZY]] of order $n$.\nIn terms of $F(x, t) = f^t(x)$, clause 3 is the [[translation equation|CPE1ANDDJ0]] $F(F(x, t), s) = F(x, s + t)$.\nIf $\\Psi \\colon X \\to \\R$ is a bijective solution of [[Schröder's equation|VTJV7JVJEA]] $\\Psi \\circ f = \\lambda \\Psi$ with $\\lambda \u003e 0$, then $f^t = \\Psi^{-1} \\circ (\\lambda^t \\Psi)$ is a fractional iteration.\nIf $A \\colon X \\to \\R$ is a bijective solution of the [[Abel equation|68SHB7Z721]] $A \\circ f = A + 1$, then $f^t = A^{-1} \\circ (A + t)$ is one.\nSome authors also call a family indexed by all real $t$, or only by rational $t$, a fractional iteration.\n",
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