Homogeneous Dynamics, Ratner Theorems, and Equidistribution¶
Job: develop the homogeneous spaces \(G/\Gamma\) and the \(H\)-actions, the Ratner orbit closure / measure classification / equidistribution for unipotent flows, the Margulis arithmeticity / the Oppenheim conjecture, the quantitative Ratner (Einsiedler–Margulis–Venkatesh), and the Dani / horocycle / \(p\)-adic — with the Oppenheim via the Ratner unipotent on \(\mathrm{SO}(2,1)/\Gamma\) as the worked example.
Core¶
Homogeneous dynamics and the Lie-group actions¶
The homogeneous space \(G/\Gamma\) (a Lie group \(G\) mod a lattice \(\Gamma\), e.g. \(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})\), \(\mathrm{SL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{Z})\) the modular / the locally symmetric spaces). The dynamics: a subgroup \(H\subset G\) acts on \(G/\Gamma\) by the left translation, the orbits \(Hx\subset G/\Gamma\).
| Homogeneous space | Lattice | \(H\)-action | Orbits |
|---|---|---|---|
| \(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})\) | \(\mathrm{SL}_2(\mathbb{Z})\) (non-cocompact) | the geodesic / horocycle | the modular surface |
| \(\mathrm{SL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{Z})\) | \(\mathrm{SL}_n(\mathbb{Z})\) (arithmetic) | the unipotent one-parameter | the locally symmetric |
| \(G/\Gamma\) | \(\Gamma\) discrete, \(\mathrm{vol}<\infty\) | \(H\subset G\) left translation | \(Hx\subset G/\Gamma\) |
The Ratner theorems (Rigidity / Measure Classification / Equidistribution)¶
Theorem (Ratner orbit closure). For a unipotent flow \(u_t=\exp(tN)\) (\(N\) nilpotent) on \(G/\Gamma\), every orbit closure \(\overline{Ux}\) is a homogeneous \(H\)-orbit (the orbit is closed in the homogeneous sub-orbit, \(Hx\) with \(H\subset G\) a closed connected subgroup, the Rigidity — the unipotent orbit closures are the algebraic / the closed).
Theorem (Ratner measure classification). The \(U\)-invariant ergodic measures on \(G/\Gamma\) are the algebraic (the Haar on the homogeneous \(H\)-orbit, the Ratner / the Dani–Margulis).
Theorem (Ratner equidistribution). The unipotent orbits equidistribute (the \(\{u_t x\}\) equidistributes in the \(H\)-orbit, the time average = the space average).
| Ratner theorem | Statement |
|---|---|
| Orbit closure | \(\overline{Ux}=Hx\) (the closed homogeneous \(H\)-orbit, the unipotent) |
| Measure classification | the \(U\)-invariant ergodic measures are the Haar on \(Hx\) (the algebraic) |
| Equidistribution | the \(\{u_t x\}\) equidistributes in \(Hx\) (the time = the space average) |
| Unipotent / algebraic | the unipotent (the \(N\) nilpotent) the algebraic orbit closures (the rigidity) |
The Margulis / arithmetic lattices and the Oppenheim conjecture¶
Theorem (Margulis arithmeticity). The irreducible lattices in the higher-rank Lie groups (\(\mathrm{rank}_\mathbb{R} G\ge2\)) are arithmetic (the Margulis arithmeticity theorem, the lattices as the arithmetic groups).
Theorem (Oppenheim conjecture, Margulis). The values of an indefinite irrational quadratic form \(Q\) on \(\mathbb{Z}^n\) (\(n\ge3\)) are dense in \(\mathbb{R}\) (the \(Q(\mathbb{Z}^n)\) dense, the Oppenheim / the Margulis via the Ratner / the unipotent dynamics).
| Margulis / Oppenheim | Statement |
|---|---|
| Margulis arithmeticity | the irreducible lattices in rank \(\ge2\) Lie groups are arithmetic |
| Oppenheim / Margulis | the \(Q(\mathbb{Z}^n)\) dense for the indefinite irrational \(Q\) (\(n\ge3\)) |
| Rank \(\ge2\) / rigidity | the higher-rank the rigidity (the arithmetic, the Margulis); the rank 1 the non-rigid |
The quantitative equidistribution and the Mozes–Shah–Einsiedler–Margulis–Venkatesh¶
| Quantitative Ratner / number theory | Statement |
|---|---|
| Quantitative Ratner (Mozes–Shah, EMV) | the effective / the quantitative equidistribution: the rate of the equidistribution of the unipotent orbits |
| Effective Oppenheim (EMV) | the quantitative Oppenheim / the rate of the density |
| LCD / Littlewood (Einsiedler–Katok–Lindenstrauss) | the \(\times2\times3\) / the Furstenberg via the homogeneous; \(\liminf n\|n\alpha\|\|n\beta\|=0\) the Littlewood |
| Number theory | the relation to the analytic number theory / equidistribution |
The Dani / horocycle and the \(p\)-adic¶
| Dani / horocycle / \(p\)-adic | Description |
|---|---|
| Dani / horocycle flows | the horocycle on \(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})\) (the Dani theorem, the horocycle orbit closures) |
| \(p\)-adic homogeneous | \(G=\mathrm{SL}_2(\mathbb{R})\times\mathrm{SL}_2(\mathbb{Q}_p)\) (the \(S\)-arithmetic, the Ratner / the \(p\)-adic) |
| Representation theory | the homogeneous / the representation theory (the Lie group / the representation) |
See it¶
Oppenheim conjecture via the Ratner unipotent dynamics on \(\mathrm{SO}(2,1)/\Gamma\). The quadratic form \(Q\) (the indefinite irrational on \(\mathbb{R}^3\), \(\mathrm{SO}(Q)\) the orthogonal group), the values \(Q(\mathbb{Z}^3)\) via the \(\mathrm{SO}(Q)\)-orbit on the lattice \(\mathrm{SO}(Q)\backslash\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})\). The unipotent flow (the \(\mathrm{SO}(Q)\) contains the unipotents — the rank-1 unipotent in \(\mathrm{SO}(2,1)\)), the Ratner orbit closure (the \(\mathrm{SO}(Q)\)-orbit closure is the homogeneous, the closed). The density: the \(Q(\mathbb{Z}^3)\) dense in \(\mathbb{R}\) (the orbit dense in \(\mathrm{SO}(Q)\backslash\mathrm{SL}_3\), the Ratner equidistribution, the Oppenheim / the Margulis). The key point: the Oppenheim (the dense values of the irrational quadratic form) is proved via the Ratner unipotent dynamics on the homogeneous space — the number-theoretic conjecture as the dynamical theorem.
Check¶
Show that the \(\mathrm{SO}(Q)\)-orbit on \(\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})\) contains unipotent elements, and explain how the Ratner theorem applies.
The indefinite quadratic form \(Q\) on \(\mathbb{R}^3\) (the signature \((2,1)\), the \(\mathrm{SO}(Q)\cong\mathrm{SO}(2,1)\)). The \(\mathrm{SO}(2,1)\) contains the unipotent one-parameter subgroups (the horospherical / the nilpotent \(N\), the \(\exp(tN)\) with \(N\) nilpotent — the unipotent from the rank-1 / the hyperbolic). The orbit of \(\mathrm{SO}(Q)\) on \(\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})\): the \(\mathrm{SO}(Q)\) acts, and the unipotent one-parameter subgroups \(U\subset\mathrm{SO}(Q)\) act on the lattice. The Ratner orbit closure theorem: the \(\overline{Ux}\) is the closed homogeneous \(H\)-orbit (\(H\supset U\), the closed). The Oppenheim: the \(Q(\mathbb{Z}^3)\) dense follows from the orbit closure / the equidistribution of the unipotent flow (the Ratner, the \(Q(\mathbb{Z}^3)\) the orbit values, the dense the equidistribution).
Explain why the Margulis arithmeticity requires the rank \(\ge2\), and what fails in rank 1.
The Margulis arithmeticity: the irreducible lattices in the higher-rank Lie groups (\(\mathrm{rank}_\mathbb{R} G\ge2\)) are arithmetic. The rank \(\ge2\) is the rigidity: the higher-rank the superrigidity (the Margulis superrigidity, the representation the rigid, the arithmeticity the consequence). In rank 1 (e.g. \(\mathrm{SL}_2(\mathbb{R})\), \(\mathrm{SO}(n,1)\), \(\mathrm{SU}(n,1)\)), the lattices need not be arithmetic (the non-arithmetic lattices exist: the non-arithmetic triangle groups in \(\mathrm{SL}_2(\mathbb{R})\), the Gromov–Piatetski-Shapiro non-arithmetic in \(\mathrm{SO}(n,1)\), the non-arithmetic in \(\mathrm{SU}(n,1)\)). The rank 1 the non-rigid: the superrigidity fails, the non-arithmetic lattices exist, the arithmeticity the rank \(\ge2\) only. So the rank \(\ge2\) the rigidity / the arithmeticity, the rank 1 the non-rigid / the non-arithmetic the exceptions.
Pitfalls¶
- The Ratner the unipotent — the semisimple / the non-unipotent flows are different (the non-unipotent the non-Ratner, the horospherical / the mixing). The Ratner is the unipotent specific (the \(N\) nilpotent), the semisimple / the diagonal flows are different.
- The Margulis arithmeticity the irreducible / the rank \(\ge2\) — the rank 1 / the reducible / the non-arithmetic the exceptions (the Margulis / the Gromov–Schoen, the non-arithmetic lattices in rank 1).
- The Oppenheim the Margulis (the Ratner / the unipotent, not the direct) — the quadratic form / the \(\mathrm{SO}(Q)\) the unipotent in the \(\mathrm{SO}(Q)\) (the rank-1 unipotent), the Ratner / the unipotent dynamics, not the direct / the elementary.
- The quantitative Ratner (the Mozes–Shah / the EMV) the effective but the rates are the polynomial / the logarithmic, not the exponential (the effective the partial — the rates the polynomial / the logarithmic, not the exponential; the effective the quantitative, the rates the sub-exponential).
- The Einsiedler–Katok–Lindenstrauss the LCD the partial (the \(\liminf\) the full measure, not the all \(\alpha,\beta\) — the Littlewood the open for the general; the EKL the full-measure, not the all).
- The Dani / horocycle the orbit closures (the Dani theorem) — the horocycle the specific (the \(\mathrm{SL}_2\), the rank 1), the Ratner the general (the unipotent, the higher-rank).
Refs¶
- Ratner, "On Raghunathan's measure conjecture" (Annals 1991); "Raghunathan's topological conjecture and distributions" (Duke 1991)
- Margulis, "Oppenheim conjecture" (1987); "Arithmeticity of irreducible lattices" (the arithmeticity theorem)
- Dani–Margulis, "Orbit closures of unipotent flows" (1990); Mozes–Shah, "On the space of ergodic invariant measures" (1995)
- Einsiedler–Margulis–Venkatesh, "Effective equidistribution for horospherical flows" (2009); Einsiedler–Katok–Lindenstrauss, "Invariant measures and the set of exceptions to the Littlewood conjecture" (Annals 2006)
Next¶
ET index · Foundations · Oseledets & Pesin · Furstenberg & Joinings · Ornstein & Isomorphism · Analytic number theory · Representation theory