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  "body": "\n## Example\n\nLet $G = \\mathrm{PSL}_2(\\R)$, the quotient of the [[special linear group|0V9PMN0RBH]] $\\mathrm{SL}_2(\\R)$ by $\\set{\\pm I}$.\nAs in [[geodesic flow on a locally symmetric space|5TKBSHY6VC]], identify $G$ with the unit tangent bundle of the [[upper half-plane|X6XGGH5FPW]] $\\mathbb{H}$.\nThen the geodesic flow is $g \\mapsto g a_t$, with $a_t$ the class of $\\operatorname{diag}(e^{t/2}, e^{-t/2})$.\nLet $h_s$ be the class of $\\begin{pmatrix} 1 \u0026 s \\\\ 0 \u0026 1 \\end{pmatrix}$.\nThe (stable) horocycle flow is $\\psi_s(g) = g h_s$, for $s \\in \\R$.\nFor a [[lattice|BTYF5Y5PFY]] $\\Gamma \\subseteq G$, it descends to $\\Gamma \\backslash G$, because left and right multiplications commute.\n\nCheck of the name and of the link with the geodesic flow.\nA matrix computation gives $a_{-t} h_s a_t = h_{e^{-t} s}$.\nHence $g h_s a_t = g a_t h_{e^{-t}s}$, so the geodesic orbits of $g$ and of $g h_s$ approach each other as $t \\to +\\infty$.\nSo the orbits of $\\psi$ are the stable leaves of the geodesic flow.\nThey project to horocycles in $\\mathbb{H}$: circles tangent to the boundary, and horizontal lines.\n\n## Remarks\n\nWhen $\\Gamma$ is cocompact, Hedlund proved that the horocycle flow on $\\Gamma \\backslash G$ is minimal: every orbit is dense.\nFurstenberg proved that it is then uniquely ergodic, with the Haar measure as the only invariant probability measure.\nThe relation $a_{-t} h_s a_t = h_{e^{-t}s}$ shows that the geodesic flow renormalises the horocycle flow.\nThe horocycle flow has zero entropy, in contrast with the geodesic flow, which is an Anosov flow.\n\n## Explanations\n\n### L1 human\n\nOn the hyperbolic plane, a horocycle is a curve that looks like a circle of infinite radius, touching the boundary at one point.\nThe horocycle flow slides a unit vector along such a curve, keeping it perpendicular to the curve.\nTwo vectors on the same horocycle, when pushed forward along geodesics, come closer and closer.\nOn a closed hyperbolic surface every horocycle orbit spreads evenly over the whole space, which makes this flow a model of rigid, slowly mixing motion.\n",
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