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  "body": "\n## Example\n\nLet $\\mathbb{T}^2 = \\R^2/\\Z^2$ be the [[torus|03NM7DZXXC]] and let $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$.\nArnold's cat map is $f(x, y) = (2x + y, \\; x + y) \\bmod 1$.\nIt is well defined because $A$ maps $\\Z^2$ to $\\Z^2$, and it is a diffeomorphism because $\\det A = 1$, so $A^{-1}$ also has integer entries.\n\nCheck that it is an [[Anosov diffeomorphism|RCQH8P6DPQ]].\nThe derivative of $f$ is $A$ at every point.\nThe characteristic polynomial is $t^2 - 3t + 1$, so the [[eigenvalues|0ZND7NQ970]] are $\\lambda_\\pm = (3 \\pm \\sqrt5)/2$, with $\\lambda_+ \\approx 2.618$ and $\\lambda_- = 1/\\lambda_+ \\approx 0.382$.\nLet $E^u$ and $E^s$ be the eigenlines for $\\lambda_+$ and $\\lambda_-$; since $A$ is symmetric they are orthogonal.\nThen $\\norm{A^n v} = \\lambda_-^n \\norm{v}$ on $E^s$ and $\\norm{A^{-n} v} = \\lambda_+^{-n}\\norm{v}$ on $E^u$, which is the required hyperbolicity with $C = 1$.\n\n## Remarks\n\nThe map preserves [[Lebesgue measure|2Z2QVWZ262]], since $\\abs{\\det A} = 1$, and it is mixing for this measure.\nIts periodic points are exactly the points with rational coordinates, so they are dense.\nThe number of fixed points of $f^n$ is $\\abs{\\det(A^n - I)} = \\lambda_+^n + \\lambda_+^{-n} - 2$; for $n = 1$ this is $1$, the origin.\nIts topological entropy is $\\log \\lambda_+$.\n\n## Explanations\n\n### L1 human\n\nDraw a picture on a square whose opposite sides are glued, so that it is really a torus.\nShear and stretch the picture by the matrix, then cut the pieces that stick out and shift them back into the square.\nAfter a few steps the picture looks like random noise, because nearby points are pulled apart by a factor of about $2.6$ each time.\nYet the map is exactly invertible, and every point with rational coordinates returns to where it started.\n",
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