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  "body": "\n## Definition\n\nLet $M$ be a [[compact|0QPHW703PV]] Riemannian manifold and let $f \\colon M \\to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].\nThen $f$ is an *Anosov diffeomorphism* if $M$ itself is a [[hyperbolic set|NY9MCZXY6N]] of $f$.\nThat is, there is a $Df$-invariant splitting $T_x M = E^s_x \\oplus E^u_x$ for all $x \\in M$, and there are constants $C \u003e 0$ and $0 \u003c \\lambda \u003c 1$.\nFor all $n \\ge 0$, they satisfy $\\norm{Df^n v} \\le C\\lambda^n \\norm{v}$ for $v \\in E^s$ and $\\norm{Df^{-n} v} \\le C\\lambda^n\\norm{v}$ for $v \\in E^u$.\n\n## Remarks\n\nThe property does not depend on the choice of Riemannian metric, since $M$ is compact.\nExample: let $A$ be a $2 \\times 2$ integer matrix with $\\det A = \\pm 1$ and no eigenvalue of modulus $1$, and let $f_A$ be the induced map of the torus $\\R^2/\\Z^2$.\nThen $Df_A = A$ at every point, and the eigenlines of $A$ give $E^s$ and $E^u$ with $C = 1$ after a linear change of metric.\nFor $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ the eigenvalues are $(3 \\pm \\sqrt5)/2$, which gives [[Arnold's cat map|BSCZN8Y1AN]].\nNon-example: for $A = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ the map $f_A$ is a diffeomorphism of the torus, but both eigenvalues equal $1$, so no direction is contracted and the condition fails.\nAll known Anosov diffeomorphisms are, up to topological conjugacy, hyperbolic automorphisms of tori, nilmanifolds or infranilmanifolds; whether this is always the case is an open question.\n",
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