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  "body": "\n## Statement\n\nLet $G$ be a connected [[Lie group|7JBRGCVJRB]], let $\\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$, and let $X = G/\\Gamma$.\nAn element $u \\in G$ is *unipotent* if $\\mathrm{Ad}(u)$ is a unipotent linear map of the Lie algebra of $G$.\nLet $H$ be a connected closed subgroup of $G$ generated by one-parameter subgroups of unipotent elements.\nA probability measure $\\mu$ on $X$ is *homogeneous* if there are a closed subgroup $L$ of $G$ and $x \\in X$ with the following properties.\nThe orbit $Lx$ is closed, and $\\mu$ is the $L$-invariant [[probability measure|059TJN0NZ1]] supported on $Lx$.\nThen:\n\n1. (Measure classification) every $H$-invariant and ergodic Borel probability measure on $X$ is homogeneous;\n2. (Orbit closures) for every $x \\in X$, the [[closure|0QEAJB91GZ]] of $Hx$ equals $Lx$ for a closed subgroup $L \\supseteq H$ such that $Lx$ carries an $L$-invariant probability measure;\n3. (Equidistribution) let $(u_t)_{t \\in \\R}$ be a one-parameter subgroup of unipotent elements, and let $x \\in X$.\n   Then there is a homogeneous measure $\\mu_x$ such that, for every bounded continuous $f$ on $X$,\n$$\\lim_{T \\to \\infty} \\frac{1}{T} \\int_0^T f(u_t x) \\, dt = \\int_X f \\, d\\mu_x.$$\n\n## Remarks\n\nMarina Ratner proved these theorems around 1990 and 1991.\nThe proof of the measure classification uses the polynomial divergence of nearby unipotent orbits: two such orbits drift apart slowly and mainly along the centraliser of the flow.\nOrbit closures and equidistribution then follow from the measure classification together with the non-divergence results of Margulis and Dani.\n\nExample: $G = \\mathrm{SL}_2(\\R)$, $\\Gamma = \\mathrm{SL}_2(\\Z)$, and $H = \\set{\\begin{pmatrix} 1 \u0026 t \\\\ 0 \u0026 1 \\end{pmatrix} : t \\in \\R}$.\nThen $X$ is the unit tangent bundle of the modular surface, and $H$ acts by the horocycle flow.\nEach $H$-orbit is either a periodic closed horocycle or dense in $X$, a fact known before Ratner; here $L = H$ or $L = G$.\n\nThe unipotent hypothesis is necessary.\nFor the diagonal subgroup of $\\mathrm{SL}_2(\\R)$, acting by the geodesic flow, there are orbits whose closures are fractal sets, not orbits of any subgroup.\n\n## Explanations\n\n### L1 human\n\nImagine a flow on a finite-volume space built from a Lie group, such as the motion of horocycles on a hyperbolic surface.\nFor unipotent flows, Ratner proved that the behaviour is always very orderly.\nEach orbit either closes up or fills out a nice smooth piece of the space evenly.\nThere are no strange fractal-shaped orbit closures, which do occur for other flows like the geodesic flow.\n\n### L2 human\n\nThese theorems are central tools of homogeneous dynamics.\nClause 2 is [[Raghunathan's conjecture|ME4ZSRAT5Z]].\nThey have applications in number theory, such as the Oppenheim conjecture on values of indefinite quadratic forms, which Margulis had proved earlier by related methods.\nThey also give counting results for integer points on homogeneous varieties.\n",
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