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  "body": "\n## Definition\n\nLet $M$ be a [[Riemannian manifold|5HDG31YEPW]] with norm $\\norm{\\cdot}$ on each tangent space, and let $f \\colon M \\to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]] with [[differential|565B0Y76XE]] $Df$.\nA set $\\Lambda \\subseteq M$ is a *hyperbolic set* of $f$ if the following hold.\n\n1. $\\Lambda$ is [[compact|0QPHW703PV]] and $f(\\Lambda) = \\Lambda$.\n2. For each $x \\in \\Lambda$ there is a splitting $T_x M = E^s_x \\oplus E^u_x$ with $Df_x(E^s_x) = E^s_{f(x)}$ and $Df_x(E^u_x) = E^u_{f(x)}$.\n3. There are constants $C \u003e 0$ and $0 \u003c \\lambda \u003c 1$ such that, for all $x \\in \\Lambda$ and $n \\ge 0$, $\\norm{Df^n_x v} \\le C \\lambda^n \\norm{v}$ for $v \\in E^s_x$ and $\\norm{Df^{-n}_x v} \\le C \\lambda^n \\norm{v}$ for $v \\in E^u_x$.\n\n## Remarks\n\nThe subspaces $E^s_x$ and $E^u_x$ are then unique and depend continuously on $x$.\nChanging the Riemannian metric changes $C$ but not the property, because $\\Lambda$ is compact; an adapted metric allows $C = 1$.\nExample: a fixed point $p$ of $f$ is a hyperbolic set exactly when no eigenvalue of $Df_p$ has modulus $1$.\nFor $f(x, y) = (x/2, 2y)$ on $\\R^2$ and $\\Lambda = \\set{0}$, take $E^s$ the $x$-axis, $E^u$ the $y$-axis, $C = 1$ and $\\lambda = 1/2$.\nNon-example: for the rotation $f(z) = e^{i\\theta} z$ of $\\R^2 = \\C$, the set $\\set{0}$ is compact and invariant, but $Df_0$ preserves lengths, so no splitting satisfies clause 3.\nA flow version replaces clause 2 by $T_x M = E^s_x \\oplus E^0_x \\oplus E^u_x$, where $E^0_x$ is spanned by the vector field, as in the definition of an [[Anosov flow|EN4A9MM7RP]].\n",
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