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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 *topological conjugacy* from $f$ to $g$ is a [[homeomorphism|0QT00PB5G2]] $h \\colon X \\to Y$ such that\n$$h \\circ f = g \\circ h,$$\nwhere $\\circ$ is [[composition|0THBK9V825]].\nThe maps $f$ and $g$ are *topologically conjugate* if such an $h$ exists.\n\nFor flows $\\varphi$ on $X$ and $\\psi$ on $Y$, a topological conjugacy is a homeomorphism $h \\colon X \\to Y$ with $h \\circ \\varphi_t = \\psi_t \\circ h$ for all $t \\in \\R$.\n\n## Remarks\n\nThe relation $h \\circ f = g \\circ h$ gives $h \\circ f^n = g^n \\circ h$ for all $n \\ge 0$, so $h$ maps orbits of $f$ onto orbits of $g$.\nIn particular $h$ maps fixed points to fixed points and periodic points of minimal period $p$ to periodic points of minimal period $p$.\nTopological conjugacy is an equivalence relation on continuous self-maps.\n\nExample: the tent map $T(x) = 1 - \\abs{2x - 1}$ and the logistic map $f(x) = 4x(1 - x)$ on $[0, 1]$ are conjugate through $h(x) = \\sin^2(\\pi x / 2)$.\nThe map $h$ is an increasing homeomorphism of $[0, 1]$.\nOne has $f(h(x)) = 4 \\sin^2(\\pi x/2) \\cos^2(\\pi x/2) = \\sin^2(\\pi x)$.\nAlso $h(T(x)) = \\sin^2(\\pi x)$ for $x \\le 1/2$, and $h(T(x)) = \\sin^2(\\pi - \\pi x) = \\sin^2(\\pi x)$ for $x \\ge 1/2$.\n\nNon-example: $f(x) = x/2$ and $g(x) = -x/2$ on $\\R$ are not conjugate.\nA conjugacy $h$ would map the only fixed point $0$ to $0$, so it would map $(0, \\infty)$ onto $(0, \\infty)$ or onto $(-\\infty, 0)$.\nSince $f$ maps $(0, \\infty)$ into itself while $g$ maps each half-line into the other one, $h \\circ f = g \\circ h$ fails.\n\nFor flows, the weaker notion of topological equivalence only asks that $h$ map orbits onto orbits and preserve their direction, not the time parametrisation.\n",
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