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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], and let $f \\colon X \\to X$ and $g \\colon Y \\to Y$ be [[continuous maps|0X3WGX95FE]].\nA *semiconjugacy* from $f$ to $g$ is a continuous [[surjective|01RVCMSN5N]] map $h \\colon X \\to Y$ such that\n$$h \\circ f = g \\circ h,$$\nwhere $\\circ$ is [[composition|0THBK9V825]].\nThen $h$ is called a *factor map*, the system $(Y, g)$ is a *factor* of $(X, f)$, and $(X, f)$ is an *extension* of $(Y, g)$.\n\n## Remarks\n\nSome authors use the word semiconjugacy without the surjectivity condition and say factor map only when $h$ is onto.\nIf $h$ is moreover a homeomorphism, it is a topological conjugacy.\nThe relation gives $h \\circ f^n = g^n \\circ h$, so $h$ maps the orbit of $x$ onto the orbit of $h(x)$, and periodic points of $f$ to periodic points of $g$.\n\nExample: let $\\Sigma = \\set{0, 1}^{\\N}$ with the product topology, let $\\sigma$ be the shift $(\\sigma x)_i = x_{i+1}$, and let $g(y) = 2y \\bmod 1$ on the circle $\\R / \\Z$.\nThe map $h(x) = \\sum_{i \\ge 0} x_i 2^{-i-1} \\bmod 1$ is continuous and onto, since every number in $[0, 1]$ has a binary expansion.\nOne has $h(\\sigma x) = \\sum_{i \\ge 0} x_{i+1} 2^{-i-1} = 2 h(x) - x_0$, which equals $2 h(x)$ modulo $1$, so $h \\circ \\sigma = g \\circ h$.\nThis $h$ is not injective, since $(1, 0, 0, \\dots)$ and $(0, 1, 1, \\dots)$ have the same image.\n\nNon-example: the inclusion $h \\colon \\set{0} \\to \\R$ satisfies $h \\circ f = g \\circ h$ for $f = \\id$ and $g(y) = y/2$, but it is not surjective, so it is not a factor map.\n",
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