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  "body": "\n## Definition\n\nLet $M$ be a [[compact|0QPHW703PV]] Riemannian manifold with norm $\\norm{\\cdot}$ on each tangent space.\nLet $X$ be a $C^1$ [[vector field|0N67NBKQ5T]] on $M$ with no zeros, and let $\\varphi_t$ be its [[flow|02903SSA48]].\nThe flow $\\varphi_t$ is an *Anosov flow* if the following hold.\n\n1. For each $x \\in M$ there is a splitting $T_x M = E^s_x \\oplus E^0_x \\oplus E^u_x$, where $E^0_x$ is the line spanned by $X(x)$.\n2. The splitting is invariant: $D\\varphi_t(E^s_x) = E^s_{\\varphi_t(x)}$ and $D\\varphi_t(E^u_x) = E^u_{\\varphi_t(x)}$ for all $t \\in \\R$.\n3. There are constants $C \u003e 0$ and $\\lambda \u003e 0$ such that, for all $x \\in M$ and $t \\ge 0$, $\\norm{D\\varphi_t v} \\le C e^{-\\lambda t}\\norm{v}$ for $v \\in E^s_x$ and $\\norm{D\\varphi_{-t} v} \\le C e^{-\\lambda t}\\norm{v}$ for $v \\in E^u_x$.\n\n## Remarks\n\nExample: the [[geodesic flow|VF43FTDA1H]] on the unit tangent bundle of a compact surface of constant curvature $-1$ is an Anosov flow.\nIn the model $\\Gamma \\backslash \\mathrm{PSL}_2(\\R)$, the flow is right multiplication by $a_t = \\operatorname{diag}(e^{t/2}, e^{-t/2})$, and $E^s$, $E^u$ are tangent to the orbits of the two [[horocycle flows|2DC88BMQVX]].\nConjugating $\\begin{pmatrix} 1 \u0026 s \\\\ 0 \u0026 1\\end{pmatrix}$ by $a_t$ multiplies $s$ by $e^{t}$ or by $e^{-t}$, which gives clause 3 with $\\lambda = 1$.\nAnother example is the suspension flow of an [[Anosov diffeomorphism|X1P48KQFXQ]] with constant roof function.\nNon-example: the linear flow $\\varphi_t(x) = x + t v$ on the torus $\\R^n/\\Z^n$ has derivative the identity, so no direction is contracted and clause 3 fails.\n",
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