User:Jonathan R. Williford/Ratner Theory

Ratner Theory is a theory about rigid properties of unipotent actions on homogeneous spaces of Lie groups (it is better to use the term "\(\textrm{Ad}\)-unipotent" instead of "unipotent").

The theory was first developed in Ratner (1982a), Ratner (1982b), Ratner (1983) to show that unipotent flows on homogeneous spaces of \(SL(2,{\mathbb R})\) (such flows are called horocycle flows) are rigidly linked to the algebraic structure of the underlying homogeneous space. Namely, it was shown that measure theoretic isomorphisms of horocycle flows must be algebraic and imply the isometry of the underlying surfaces. Also factors and joinings of horocycle flows were shown to be algebraic. These results for the most general case are stated in Theorems S4–S6 below.

The ideas from Ratner (1982a)–Ratner (1983) led to further far reaching development of the Ratner Theory in Ratner (1990a)–Ratner (1991b) culminated in Theorems 1–3 and S1–S3 below.

Basic results of the Ratner theory

Let \({\mathbf G}\) be a locally compact second countable topological group, \({\mathbf \Gamma}\) a discrete subgroup of \({\mathbf G}\), and \({\mathbf \Gamma}\setminus{\mathbf G} = \{{\mathbf \Gamma}{\mathbf h}: {\mathbf h} \in {\mathbf G}\}\). One denotes by \(\pi: {\mathbf G} \to {\mathbf \Gamma}\setminus{\mathbf G}\) the covering projection \(\pi({\mathbf h}) = {\mathbf \Gamma}{\mathbf h}\), \({\mathbf h} \in {\mathbf G}\). Recall that the group \({\mathbf \Gamma}\) is called a lattice in \({\mathbf G}\) if there is a finite \({\mathbf G}\)-invariant measure \(\nu_[[:Template:\mathbf G]]\) on \({\mathbf \Gamma}\setminus{\mathbf G}\). (In this case it is assumed that \(\nu_[[:Template:\mathbf G]]\) is a probability measure, i.e., \(\nu_[[:Template:\mathbf G]]({\mathbf \Gamma}\setminus{\mathbf G}) = 1\).) The group \({\mathbf G}\) acts on \({\mathbf \Gamma}\setminus{\mathbf G}\) by right translations\[x \to x{\mathbf g}\], \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\), \({\mathbf g} \in {\mathbf G}\). If \({\mathbf U}\) is a subgroup of \({\mathbf G}\) and \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\) then the set \(x{\mathbf U} = \{x{\mathbf u}: {\mathbf u} \in {\mathbf U}\}\) is called the \({\mathbf U}\)-orbit of \(x\). One studies the closures of \({\mathbf U}\)-orbits and the ergodic \({\mathbf U}\)-invariant Borel probability measures on \({\mathbf \Gamma}\setminus{\mathbf G}\).

Here are some natural examples. Suppose \({\mathbf G}\) is a real Lie group, \({\mathbf U} = \{{\mathbf u}(t): t \in {\mathbb R}\}\) a one-parameter subgroup of \({\mathbf G}\), and \(x{\mathbf U}\) a periodic orbit. Then \(x{\mathbf U} = \overline{x{\mathbf U}}\) and the normalized length measure on \(x{\mathbf U}\) is \({\mathbf U}\)-invariant and ergodic.

For a more general example suppose that the closure \(\overline{x{\mathbf U}}\) coincides with the orbit of a larger group \({\mathbf H}\) containing \({\mathbf U}\), i.e., \(\overline{x{\mathbf U}} = x{\mathbf H}\). In addition, it might happen that \(x{\mathbf H}\) is the support of an \({\mathbf H}\)-invariant Borel probability measure \(\nu_[[:Template:\mathbf H]]\) (this happens if and only if \({\mathbf x}{\mathbf H}{\mathbf x}^{-1} \cap {\mathbf \Gamma}\) is a lattice in \({\mathbf x}{\mathbf H}{\mathbf x}^{-1}\), \({\mathbf x} \in \pi^{-1}\{x\}\)) that is ergodic for the action of \({\mathbf U}\).

These examples motivate the following definitions.

Definition 1. A subset \(A \subset {\mathbf \Gamma}\setminus{\mathbf G}\) is called homogeneous if there exists a closed subgroup \({\mathbf H} \subset {\mathbf G}\) and a point \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\) such that \(A = x{\mathbf H}\) and \(x{\mathbf H}\) is the support of an \({\mathbf H}\)-invariant Borel probability measure \(\nu_[[:Template:\mathbf H]]\).

This definition of \(x{\mathbf H}\) being homogeneous is different from the commonly used one where the existence of a finite \({\mathbf H}\)-invariant measure on \(x{\mathbf H}\) is not required.

Definition 2. A Borel probability measure \(\mu\) on \({\mathbf \Gamma}\setminus{\mathbf G}\) is algebraic if there exist \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\) and a closed subgroup \({\mathbf H} \subset {\mathbf G}\) such that \(x{\mathbf H}\) is homogeneous and \(\mu = \nu_[[:Template:\mathbf H]]\).

Equivalently, \(\mu\) is algebraic if there is \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\) such that \(\mu(x{\mathbf \Lambda}(\mu)) = 1\), where \[ {\mathbf \Lambda}(\mu) = \{{\mathbf g} \in {\mathbf G}:\] the action of \({\mathbf g}\) on \({\mathbf \Gamma}\setminus{\mathbf G}\) preserves \(\mu\}. \) This terminology of a "homogeneous" set and an "algebraic" measure was introduced in Ratner (1990a).

For a subgroup \({\mathbf U} \subset {\mathbf G}\) it is rather exceptional to have all orbit closures homogeneous or all ergodic invariant measures algebraic. However, there are some \({\mathbf U}\) for which this happens. To characterize these \({\mathbf U}\) one needs the following definitions.

Let \({\mathbf G}\) be a Lie group over a field \(\kappa\) (where \(\kappa\) is either the real field or a \(p\)-adic field) with the Lie algebra \({\mathfrak G}\). For \({\mathbf g} \in {\mathbf G}\) let \(\textrm{Ad}_[[:Template:\mathbf g]]: {\mathfrak G} \to {\mathfrak G}\) denote the differential at the identity of the map \({\mathbf h} \to {\mathbf g}^{-1}{\mathbf h}{\mathbf g}\), \({\mathbf h} \in {\mathbf G}\). Then \(\textrm{Ad}_[[:Template:\mathbf g]]\) (called the adjoint map of \({\mathbf g}\)) is a linear automorphism of \({\mathfrak G}\).

It is a fact that there is a neighborhood \({\mathfrak O}\) of zero in \({\mathfrak G}\) such that the exponential map \(\exp: {\mathfrak O} \to {\mathbf G}\) is well defined on \({\mathfrak O}\) and maps \({\mathfrak O}\) diffeomorphically onto a neighborhood of the identity \({\mathbf e}\) in \({\mathbf G}\). (When \(\kappa = {\mathbb R}\) the map \(\exp\) is defined on all of \({\mathfrak G}\).) If \({\mathbf x},{\mathbf y} \in {\mathbf G}\) and \({\mathbf y} = {\mathbf x} \exp v\) for some \(v \in {\mathfrak O}\) with \(\textrm{Ad}_{{\mathbf g}^r}(v) \in {\mathfrak O}\) for all \(r = 1,\dots,n\) and some \(0 \le n \in {\mathbb Z}\) then \({\mathbf y}{\mathbf g}^r = {\mathbf x}{\mathbf g}^r \exp(\textrm{Ad}_{{\mathbf g}^r}(v))\) for all \(r = 1,\dots,n\). Thus \(\textrm{Ad}_{{\mathbf g}^r}\) characterizes the divergence of \({\mathbf y}{\mathbf g}^r\) from \({\mathbf x}{\mathbf g}^r\) when \(r\) runs from \(1\) to \(n\).

An element \({\mathbf u} \in {\mathbf G}\) is called \(\textrm{Ad}\)-unipotent if \(\textrm{Ad}_[[:Template:\mathbf u]]\) is a unipotent element of \(GL({\mathfrak n},\kappa)\), \({\mathfrak n} = \dim {\mathfrak G}\), i.e., every eigenvalue of \(\textrm{Ad}_[[:Template:\mathbf u]]\) equals one. Then \(\textrm{Ad}_{{\mathbf u}^r} = \sum_{k=0}^m(r^kT_[[:Template:\mathbf u]]^k)/k!\) for all \(r \in {\mathbb Z}\) and some integer \(m \ge 0\), where \(T_[[:Template:\mathbf u]]\) is a nilpotent endomorphism of \({\mathfrak G}\). This polynomial (in \(r\)) form of \(\textrm{Ad}_{{\mathbf u}^r}\) plays a crucial role in all of the results stated below. It shows that \(\textrm{Ad}\)-unipotent orbits diverge polynomially.

The following Theorems 1–3 stated for real Lie groups represent the central results of the Ratner Theory.

Theorem 1 [Ratner (1990a), Ratner (1990b), Ratner (1991a)] (Measure rigidity for groups generated by \(\textrm{Ad}\)-unipotent elements). Let \({\mathbf G}\) be a connected Lie group and \({\mathbf U}\) a connected subgroup of \({\mathbf G}\) generated by \(\textrm{Ad}\)-unipotent elements of \({\mathbf G}\). Then given any discrete subgroup \({\mathbf \Gamma}\) of \({\mathbf G}\) every ergodic \({\mathbf U}\)-invariant Borel probability measure on \({\mathbf \Gamma}\setminus{\mathbf G}\) is algebraic.

Theorem 2 [Ratner (1991b)] (Orbit closures for groups generated by \(\textrm{Ad}\)-unipotent elements). Let \({\mathbf G}\) and \({\mathbf U}\) be as in Theorem 1. Then given any lattice \({\mathbf \Gamma}\) in \({\mathbf G}\) and any \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\) the closure of the orbit \(x{\mathbf U}\) in \({\mathbf \Gamma}\setminus{\mathbf G}\) is homogeneous.

The term "measure rigidity" was introduced in Ratner (1990a). Note that every connected semisimple \({\mathbf U} \subset {\mathbf G}\) without compact factors is generated by \(\textrm{Ad}\)-unipotent elements of \({\mathbf G}\).

To state Theorem 3 one needs the following.

Definition 3. Let \({\mathbf U} = \{{\mathbf u}(t): t \in {\mathbb R}\}\) be an arbitrary one-parameter subgroup of \({\mathbf G}\). A point \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\) is called generic for \({\mathbf U}\) if there exists a closed subgroup \({\mathbf H} \subset {\mathbf G}\) such that \(\overline{x{\mathbf U}} = x{\mathbf H}\) is homogeneous and \(\frac {1}{t} \int_0^t f(x{\mathbf u}(s))ds \to \int_{{\mathbf \Gamma}\setminus{\mathbf G}} f\ d\nu_[[:Template:\mathbf H]]\) as \(t \to \infty\), for every bounded continuous function \(f\) on \({\mathbf \Gamma}\setminus{\mathbf G}\).

A similar definition can be given for a one-generator \({\mathbf U} = \{{\mathbf u}^k: k \in {\mathbb Z}\}\) replacing the integral by the sum \(\sum_{k=0}^{n-1} f(x{\mathbf u}^k)/n\).

Theorem 3 [Ratner (1991b)] (Uniform distribution of \(\textrm{Ad}\)-unipotent flows). Let \({\mathbf G}\) be a connected Lie group and \({\mathbf U}\) a one-parameter or one-generator \(\textrm{Ad}\)-unipotent subgroup of \({\mathbf G}\). Then given any lattice \({\mathbf \Gamma}\) of \({\mathbf G}\) every point \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\) is generic for \({\mathbf U}\) and \({\mathbf U}\) acts ergodically on \((\overline{x{\mathbf U}} = x{\mathbf H},\nu_[[:Template:\mathbf H]])\).

The ideas and methods developed in the proofs of Theorems 1–3 were used to extend the Theorems to \(p\)-adic Lie groups and, more generally, to Cartesian products of real and \(p\)-adic Lie groups, the so-called \(S\)-arithmetic setting.

More specifically, let \({\mathbb S}\) be a finite set and for each \(s \in {\mathbb S}\) let \({\mathbb Q}_s\) be either the real field \({\mathbb R}\) or the field of \(p_s\)-adic numbers for some prime \(p_s\). In the latter case we call \(s\) ultrametric, otherwise \(s\) is called real.

For \(s \in {\mathbb S}\) let \({\mathbf G}_s\) be a Lie group over \({\mathbb Q}_s\) with the Lie algebra \({\mathfrak G}_s\) and let \({\mathbf G}_[[:Template:\mathbb S]] = \prod \{{\mathbf G}_s \in s \in {\mathbb S}\}\) denote the Cartesian product of \({\mathbf G}_s\), \(s \in {\mathbb S}\).

Let \(\eta: {\mathbf G}_s \to {\mathbf G}_[[:Template:\mathbb S]]\) denote the natural embedding of \({\mathbf G}_s\) in \({\mathbf G}_[[:Template:\mathbb S]]\) and let \({\mathbf U}_s = \{{\mathbf u}_s(t): t \in {\mathbb Q}_s\}\) be a one-parameter \(\textrm{Ad}\)-unipotent subgroup of \({\mathbf G}_s\). Then \({\mathbf U} = \eta({\mathbf U}_s) = \{{\mathbf u}(t) = \eta({\mathbf u}_s(t)): t \in {\mathbb Q}_s\}\) is called a one-parameter \(\textrm{Ad}\)-unipotent subgroup of \({\mathbf G}_[[:Template:\mathbb S]]\).

For an ultrametric \(s \in {\mathbb S}\) we call \({\mathbf G}_s\) regular if \(\ker \textrm{Ad}_{{\mathbf G}_s}\) is the center of \({\mathbf G}_s\) and the orders of all finite subgroups of \({\mathbf G}_s\) do not exceed a constant depending only on \({\mathbf G}_s\). It is a fact that if \(\kappa\) is a finite extension of \({\mathbb Q}_s\) with an ultrametric \(s\) then \(GL({\mathfrak n},\kappa)\), \({\mathfrak n} \in {\mathbb Z}^+\), and its Zariski closed and connected subgroups (viewed as Lie groups over \({\mathbb Q}_s\)) are regular.

The following theorems extend Theorems 1–3 to the \(S\)-arithmetic setting. In these theorems it is assumed that when \(s \in {\mathbb S}\) is ultrametric, the group \({\mathbf G}_s\) is regular.

Theorem S1 (Measure Rigidity [Ratner (1993)]). Let \({\mathbf H}\) be a closed subgroup of \({\mathbf G}_[[:Template:\mathbb S]]\) and \({\mathbf U}\) a subgroup of \({\mathbf H}\) generated by one-parameter \(\textrm{Ad}\)-unipotent subgroups of \({\mathbf G}_[[:Template:\mathbb S]]\). Then given any discrete subgroup \({\mathbf \Gamma}\) of \({\mathbf H}\) every ergodic \({\mathbf U}\)-invariant Borel probability measure on \({\mathbf \Gamma}\setminus{\mathbf H}\) is algebraic.

Theorem S2 (Orbit Closures [Ratner (1993)]). Let \({\mathbf H}\) and \({\mathbf U}\) be as in Theorem  S1. Then given any lattice \({\mathbf \Gamma}\) of \({\mathbf H}\) and any \(x \in {\mathbf \Gamma}\setminus{\mathbf H}\) the closure \(\overline{x{\mathbf U}}\) of the orbit \(x{\mathbf U}\) in \({\mathbf \Gamma}\setminus{\mathbf H}\) is homogeneous.

To state Theorem S3 one needs the following notations.

Let \({\mathbf H}\) be a closed subgroup of \({\mathbf G}_[[:Template:\mathbb S]]\), \({\mathbf \Gamma}\) a discrete subgroup of \({\mathbf H}\), and \({\mathbf U} = \eta({\mathbf U}_s) = \{{\mathbf u}(t): t \in {\mathbb Q}_s\}\), \(s \in {\mathbb S}\), a one-parameter \(\textrm{Ad}\)-unipotent subgroup of \({\mathbf G}_[[:Template:\mathbb S]]\) contained in \({\mathbf H}\). For \(\tau > 0\) let \[ F_s(\tau) = \{t \in {\mathbb Q}_s: |t|_s \le \tau\}, \] where \(|\cdot|_s\) denotes the normalized absolute value on \({\mathbb Q}_s\). When \(s\) is ultrametric, \(F_s(\tau)\) is a compact open subgroup of \({\mathbb Q}_s\). One denotes by \(\lambda_s\) a Haar measure on \({\mathbb Q}_s\).

Theorem S3 (Uniform Distribution [Ratner (1993)]). Given any lattice \({\mathbf \Gamma}\) of \({\mathbf H}\) and any \(x \in {\mathbf \Gamma}\setminus{\mathbf H}\) there exists a closed subgroup \({\mathbf L}\) of \({\mathbf H}\) such that \(\overline{x{\mathbf U}} = x{\mathbf L}\) is homogeneous, \({\mathbf U}\) acts ergodically on \((\overline{x{\mathbf U}} = x{\mathbf L},\nu_[[:Template:\mathbf L]])\), and \[ \frac {1}{\lambda_s(F_s(\tau))} \int_{F_s(\tau)} f(x{\mathbf u}(t))d\lambda_s(t) \to \int_{{\mathbf \Gamma}\setminus{\mathbf H}} f\ d\nu_[[:Template:\mathbf L]]\] as \(\tau \to \infty, \) for every bounded continuous function \(f\) on \({\mathbf \Gamma}\setminus{\mathbf H}\).

Theorem S3 holds also for \({\mathbf U}\) being a one-parameter \(\textrm{Ad}\)-unipotent subgroup of a Lie group \({\mathbf G}\) over a finite extension of a \(p\)-adic field \({\mathbb Q}_p\).

Theorem S1 implies important ergodic-theoretic results (Theorems S4–S6 below) extending rigidity theorems of Ratner (1982a)–Ratner (1983) to the \(S\)-arithmetic setting.

Indeed, let \({\mathbf G}_i = {\mathbf G}_{{\mathbb S}_i} = \sqcap_{\sigma \in {\mathbb S}_i} {\mathbf G}_{\sigma}^{(i)}\) and \({\mathbf \Gamma}_i\) a lattice in \({\mathbf G}_i\), \(i = 1,2\). Suppose \({\mathbb S}_1 \cap {\mathbb S}_2 \ne \emptyset\) and let \(\sigma \in {\mathbb S}_1 \cap {\mathbb S}_2\). Let \({\mathbf U}_i = \{{\mathbf u}_i(s): s \in {\mathbb Q}_{\sigma}\}\) be a one-parameter \(\textrm{Ad}\)-unipotent subgroup of \({\mathbf G}_i\) lifted from \({\mathbf G}_{\sigma}^{(i)}\), \(i = 1,2\) and let \(\mu_i\) denote the \({\mathbf G}_i\)-invariant probability measure on \(X_i = {\mathbf \Gamma}_i\setminus{\mathbf G}_i\), \(i = 1,2\).

Theorem S4 (The Joinings Theorem [Ratner (1998)]). Let \(\mu\) be an ergodic joining of the action of \({\mathbf U}_1\) on \((X_1,\mu_1)\) and of \({\mathbf U}_2\) on \((X_2,\mu_2)\). Then \(\mu\) is algebraic. If \({\mathbf G}_1\) and \({\mathbf G}_2\) are simple and \(\mu\) is nontrivial then \({\mathbf G}_1\) and \({\mathbf G}_2\) are locally isomorphic.

Theorem S5 (The Rigidity Theorem [Ratner (1998)]). Assume that for every open subgroup \({\mathbf H}_i \subset {\mathbf G}_i\) with \({\mathbf U}_i \subset {\mathbf H}_i\), \(\mu_i({\mathbf e}_i{\mathbf H}_i) = 1\), the group \({\hat {\mathbf \Gamma}}_i = {\mathbf H}_i \cap {\mathbf \Gamma}_i\) contains no nontrivial normal subgroups of \({\mathbf H}_i\), \(i = 1,2\). Suppose that \({\mathbf U}_i\) acts ergodically on \((X_i,\mu_i)\), \(i = 1,2\) and there is a measure preserving map \(\psi: (X_1,\mu_i) \to (X_2,\mu_2)\) such that \(\psi(x{\mathbf u}_1(s)) = \psi(x){\mathbf u}_2(s)\) for \(\mu_1\)-a.e. \(x \in X_1\) and all \(s \in {\mathbb Q}_{\sigma}\). Then there is an open subgroup \({\mathbf H}_i \subset {\mathbf G}_i\) with \({\mathbf U}_i \subset {\mathbf H}_i\), \(\mu_i(e{\mathbf H}_i) = 1\), \(i = 1,2\), an element \({\mathbf c} \in {\mathbf H}_2\) and a continuous surjective homomorphism \(\alpha: {\mathbf H}_1 \to {\mathbf H}_2\) such that \(\alpha({\hat {\mathbf \Gamma}}_1) \subset {\mathbf c}^{-1}{\hat {\mathbf \Gamma}}_2{\mathbf c}\) and \(\psi({\mathbf \Gamma}_1{\mathbf h}) = {\mathbf \Gamma}_2{\mathbf c}\alpha({\hat {\mathbf h}})\) for \(\mu_1\)-a.e.  \({\mathbf \Gamma}_1{\mathbf h} \in X_1\), where \({\hat {\mathbf h}} \in {\mathbf H}_1\), \({\mathbf \Gamma}_1{\mathbf h} = {\mathbf \Gamma}_1{\hat {\mathbf h}}\). Also \(\alpha\) is a local isomorphism whenever \(\psi\) is finite to one or \({\mathbf H}_1\) is simple and it is an isomorphism whenever \(\psi\) is one-to-one or \({\mathbf H}_1\) is simple with trivial center.

Theorems S4, S5 hold also for \({\mathbf U}_1 = {\mathbf U}_2 = {\mathbf U}\) generated by one-parameter \(\textrm{Ad}\)-unipotent subgroups of \({\mathbf G}_i\), \(i = 1,2\).

Theorem S6 (The Factor Theorem [Ratner (1998)]). Let \({\mathbf G}_[[:Template:\mathbb S]]\) be as in Theorem  S1, \({\mathbf \Gamma}\) a lattice in \({\mathbf G}_[[:Template:\mathbb S]]\) and \({\mathbf U}\) a subgroup of \({\mathbf G}_[[:Template:\mathbb S]]\) generated by one-prameter \(\textrm{Ad}\)-unipotent subgroups of \({\mathbf G}_[[:Template:\mathbb S]]\). Suppose \({\mathbf U}\) acts ergodically on \(({\mathbf \Gamma}\setminus{\mathbf G}_[[:Template:\mathbb S]],\nu_{{\mathbf G}_[[:Template:\mathbb S]]})\) an let \(T\) be a factor of this action. Then there exists an open subgroup \({\mathbf H} \subset {\mathbf G}_[[:Template:\mathbb S]]\) containing \({\mathbf U}\) with \(\mu_{{\mathbf G}_[[:Template:\mathbb S]]}(e_[[:Template:\mathbb S]]{\mathbf H}) = 1\) such that \(T\) is isomorphic to an algebraic factor of the action of \({\mathbf U}\) on \(({\mathbf \Gamma} \cap {\mathbf H}\setminus{\mathbf H},\nu_[[:Template:\mathbf H]])\).

Theorems 1 and 2 were extended to some subgroups \({\mathbf U}\) of \({\mathbf G}\) not generated by unipotent elements. For example, the theorems hold for \({\mathbf U}\) being a parabolic subgroup of a connected semisimple subgroup of \({\mathbf G}\) without compact factors, and more generally , for subgroups \({\mathbf U}\) of \({\mathbf G}\) epimorphic to connected semisimple subgroups of \({\mathbf G}\) without compact factors (see Mozes (1995), Shah and Weiss (2000), Ratner ( 1994a), Ratner ( 1998)). Also Theorem 3 was extended in Shah (1994) to more general polynomial actions.

Also a number of important results were obtained in Einsiedler and Katok (2005); Einsiedler, Katok and Lindenstrauss (to appear); and Einsiedler and Lindenstrauss (2005)–Einsiedler and Lindenstrauss (to appear) extending Measure Rigidity Theorems 1 and S1 and Joining-Factor Theorems S4, S6 to higher rank diagonalizable actions satisfying certain positive entropy conditions. These results use Theorem S1 and the \(H\)-principle discussed below.

Basic ideas and methods of the Ratner theory

The crucial role in the Ratner Theory is played by two fundamental principles: the \(H\)-principle and the \(R\)-principle. The former was introduced and developed in Ratner (1982a)–Ratner (1983) and in a more general form in Witte (1985) and Ratner (1991b). The latter was developed in Ratner (1990a) and used in Ratner (1990a)–Ratner (1991a), Ratner (1995).

The \(H\)-principle consists of a special shearing property of \(\textrm{Ad}\)-unipotent orbits (which Ratner called the \(H\)-property in honor of the horocycle flow) and the way of utilizing it via the pointwise ergodic theorem, the polynomial divergence and the choice of appropriate compact subsets of \(M = {\mathbf \Gamma}\setminus{\mathbf G}\).

The \(H\)-property says, roughly speaking, that if \({\mathbf U} = \{{\mathbf u}(t): t \in {\mathbb R}\}\) is an \(\textrm{Ad}\)-unipotent one-parameter subgroup of a Lie group \({\mathbf G}\) and \(x,y\) are two nearby points in \(M = {\mathbf \Gamma}\setminus{\mathbf G}\) with \(y \notin x{\mathbf C}_[[:Template:\mathbf G]]({\mathbf U})\) (here \({\mathbf C}_[[:Template:\mathbf G]]({\mathbf U})\) denotes the centralizer of \({\mathbf U}\) in \({\mathbf G}\)) then the fastest divergence (shearing) of \(x{\mathbf u}(t)\) and \(y{\mathbf u}(t)\) is along \({\mathbf C}_[[:Template:\mathbf G]]({\mathbf U})\) and this divergence is polynomial in \(t\). This means that there is \({\mathbf c}(t) \in {\mathbf C}_[[:Template:\mathbf G]]({\mathbf U})\) such that the distance between \(y{\mathbf u}(t)\) and \(x{\mathbf u}(t){\mathbf c}(t)\) is small comparing to the size of \({\mathbf c}(t)\) and both the distance and the size grow polynomially in \(t\) with the latter polynomial having a larger degree.

This implies that there exists a compact subset \({\mathbf C} \subset {\mathbf C}_[[:Template:\mathbf G]]({\mathbf U})\) not containing the identity element \({\mathbf e} \in {\mathbf G}\) and \(0 < \eta < 1\) such that for any small \(\epsilon > 0\) there are \(\delta > 0\) and a large \(T > 1\) such that if \(d(x,y) < \delta\) then \(d(y{\mathbf u}(t),x{\mathbf u}(t){\mathbf c}(t)) < \epsilon\) for all \(t \in [(1-\eta)T,T]\) and some \({\mathbf c}(t) \in {\mathbf C}\).

Now let \(K\) be a compact subset of \(M\) such that the proportion of those \(t \in [0,T]\) for which \(x{\mathbf u}(t) \in K\), \(y{\mathbf u}(t) \in K\) is much closer to \(1\) than \(1-\eta\). Then there is \(t \in [(1-\eta)T,T]\) such that \(x{\mathbf u}(t) \in K\) and \(y{\mathbf u}(t) \in K\) and hence \(d(K,K{\mathbf c}(t)) < \epsilon\) for some \({\mathbf c}(t) \in {\mathbf C}\).

Now if \(\mu\) is an ergodic \({\mathbf U}\)-invariant Borel probability measure on \(M\) and \(0 \le \mu(M/K) \ll \eta\) then by ergodicity the set of those \(x \in M\) for which the proportion of \(t \in [0,T]\) with \(x{\mathbf u}(t) \in K\) is much closer to \(1\) than \(1-\eta\) has almost full \(\mu\)-measure. This allows to choose \(x,y\) as above for any \(\epsilon \to 0\) (unless \(\mu(x{\mathbf C}_[[:Template:\mathbf G]]({\mathbf U})) = 1\) for some \(x \in M\)) and hence \(K \cap K{\mathbf c} \ne \emptyset\) for some \({\mathbf c} \in {\mathbf C}\).

The compact sets \({\mathbf C}\) and \(K\) can be chosen to suit various purposes. For example, choosing an appropriate \(K\) one can conclude from the relation \(K \cap K{\mathbf c} \ne \emptyset\), \({\mathbf c} \in {\mathbf C}\) that \({\mathbf c}\) preserves \(\mu\). One can slightly refine the \(H\)-principle by considering the divergence of \({\mathbf U}\)-orbits in a direction transversal to \({\mathbf U}\). The polynomial form of \(\textrm{Ad}_{{\mathbf u}(t)}(v)\) then shows that the fastest transverse divergence is in the direction of the normalizer \({\mathbf N}_{\mathbf G}({\mathbf U})\) of \({\mathbf U}\). Then an argument as above shows that either \(\mu(x{\mathbf U}) = 1\) for some \(x \in M\) or \(\mu\) is preserved by an element of \({\mathbf N}_{\mathbf G}({\mathbf U})\) not belonging to \({\mathbf U}\) (see Witte–Morris (2005) for details).

The \(R\)-principle generalizes the latter refinement to multidimensional \(\textrm{Ad}\)-unipotent \({\mathbf U}\) by introducing for such \({\mathbf U}\) a transversal shearing property (which Ratner called the \(R\)-property) and a similar way of utilizing it. The \(R\)-property is applied to any connected simply connected \(\textrm{Ad}\)-unipotent subgroup \({\mathbf U} \subset {\mathbf G}\). When \(\dim {\mathbf U} > 1\) one replaces the orbit intervals as above by rectangular Folner subsets of \({\mathbf U}\).

The \(R\)-property states roughly speaking that there is a compact subset \({\mathbf D}\) of the normalizer \({\mathbf N}_[[:Template:\mathbf G]]({\mathbf U})\) transverse to \({\mathbf U}\) with \({\mathbf e} \notin {\mathbf D}\) and \(0 < \eta < 1\) such that for any small \(\epsilon > 0\) there is \(\delta > 0\) and a large Folner rectangular subset \({\mathbf F} \subset {\mathbf U}\) such that if \(d(x,y) < \delta\), \(y \notin x{\mathbf N}_[[:Template:\mathbf G]]({\mathbf U})\), \(x,y \in M\) then \(d(y{\mathbf u},x{\mathbf u}{\mathbf n}({\mathbf u})) < \epsilon\) for some \({\mathbf n}({\mathbf u}) \in {\mathbf D}\) and all \({\mathbf u} \in {\mathbf A} \subset {\mathbf F}\) where \({\mathbf A}\) is a subset of \({\mathbf F}\) with \(\lambda({\mathbf A}) > \eta\lambda({\mathbf F})\) and \(\lambda\) denotes a Haar measure on \({\mathbf U}\).

The \(R\)-principle was used in the proofs of Theorems 1 and S1 in Ratner (1990a)–Ratner (1991a) and Ratner (1995). Theorems 2 and S2 follow from Theorems 3 and S3 respectively.

The proof of Theorem 3 uses Theorem 1 and the following important Theorem proved in Ratner (1991a) ( see also Shah ( 1991))

The Countability Theorem. Let \({\mathbf G}\) be a connected Lie group, \({\mathbf \Gamma}\) a discrete subgroup of \({\mathbf G}\), and \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\). Let \({\mathcal A}_x\) denote the set of all closed connected \({\mathbf H} \subset {\mathbf G}\) such that \(x{\mathbf H}\) is homogeneous and there is a connected subgroup \({\mathbf U} \subset {\mathbf H}\) generated by \(\textrm{Ad}\)-unipotent elements of \({\mathbf G}\) acting ergodically on \((x{\mathbf H},\nu_[[:Template:\mathbf H]])\). Then \({\mathcal A}_x\) is countable.

This theorem was generalized for the \(S\)-arithmetic case in Ratner (1995) and used in the proof of Theorem S3.

A linearized version of the proof of Theorem 3 was given in Dani and Margulis (1993) where it was shown that the convergence in Theorem 3 is uniform on compact subsets of \({\mathbf \Gamma}\setminus{\mathbf G}\).

Applications of Ratner theory

Theorems 1–3, S1–S3 and the methods of the Ratner Theory have been widely applied to various problems in number theory, ergodic theory, the theory of discrete groups, harmonic analysis and more. Some of these applications are discussed below.

One of the most profound results in this direction is Margulis' proof of the Oppenheim Conjecture contained in the following theorem.

Theorem A1 (Margulis (1989)). Let \(Q\) be a real, indefinite, nondegenerate quadratic form in \(n \ge 3\) variables. If \(Q\) is not a scalar multiple of a form with integer coefficients, then \(Q({\mathbb Z}^n)\) is dense in \({\mathbb R}\).

It was observed by M. S. Raghunathan that in order to prove this theorem one needs to prove Theorem 2 for \({\mathbf G} = SL(2,{\mathbb R})\), \({\mathbf \Gamma} = SL(3,{\mathbb Z})\) and \({\mathbf U} = SO(2,1)\). This was exactly what Margulis did. Subsequently it was shown in Dani and Margulis (1989) and Dani and Margulis (1990b) that the values of \(Q\) at the primitive elements of \({\mathbb Z}^n\) are dense in \({\mathbb R}\).

Using Theorem 3 and linearization method of Dani and Margulis (1993) it was possible to give quantitative versions of Margulis' Theorem A1.

Theorem A2 (Eskin, Margulis and Mozes (1998)). Let \(Q\) be as in Theorem  A1. Suppose that the signature \((p,q)\) of \(Q\) satisfies \(p \ge 3\), \(q \ge 1\). Then for any interval \((a,b)\) in \({\mathbb R}\), one has \[ \lim_{N \to \infty} \frac {\#\{v \in {\mathbb Z}^{p+q}: a < Q(v) < b,\ \|v\| \le N\}}{\textrm{vol}\{v \in {\mathbb R}^{p+q}: a < Q(v) < b,\ \|v\| \le N\}} = 1. \]

Using Theorem S2 in Borel and Prasad (1992) Theorem A1 was extended to quadratic forms over local fields of characteristic zero.

Theorem 1, the Countability Theorem and the linearization method of Dani and Margulis (1993) were used in Eskin, Mozes and Shah (1996) for the problem of counting integral points on homogeneous varieties (see also Eskin and McMullen (1993)).

Theorem S1 and the \(H\)-principle were used by E. Lindenstrauss to prove an important special case of the Quantum Unique Ergodicity Conjecture of P. Sarnak and Z. Rudnick.

Suppose \({\mathbf \Gamma}\) is a lattice in \({\mathbf G} = SL(2,{\mathbb R})\) such that \({\mathbf \Gamma}\setminus{\mathbf G}\) is compact. Then \({\mathbf \Gamma}\) acts discontinuously on the upper half complex plane \({\mathfrak H} = \{z: \textrm{Im} z > 0\}\) by linear fractional transformations and the hyperbolic metric on \({\mathfrak H}\) yields a Riemannian metric on the compact manifold \(M = {\mathbf \Gamma}\setminus{\mathfrak H}\). Let \(\textrm{vol}\) denote the normalized Riemannian volume on \(M\).

Theorem A3 (Lindenstrauss (2006)). Let \({\mathbf \Gamma}\) be a congruence lattice in \({\mathbf G}\) and let \(\varphi_n\) be a complete orthonormal sequence of eigenfunctions of the Laplacian on \(M\). Then the probability measures \(d\mu_n(x) = |\varphi_n(x)|^2d\textrm{vol}\) tend in the weak star topology to the uniform measure \(d\textrm{vol}\) on \(M\).

Theorem 1 and the \(H\)-principle were also used in Einsiedler, Katok and Lindenstrauss (to appear) to give a partial solution to the Littlewood Conjecture stating that for any \((u,v) \in {\mathbb R}^2\) one has \(\liminf_{n \to \infty} n \langle nu\rangle\langle nv\rangle = 0\), where \(\langle w\rangle\) denotes the distance of \(w \in {\mathbb R}\) to the nearest integer.

Theorem A4 (Einsiedler, Katok and Lindenstrauss (to appear)). Let \(E = \{(u,v) \in {\mathbb R}^2: \liminf_{n \to \infty} n\langle nu\rangle\langle nw\rangle > 0\}\). Then the Hausdorff dimension \(\dim_H E = 0\).

Another profound application of Ratner Theory was given by V. Vatsal (see Vatsal (2002)–Vatsal (2006)) in his study on non-vanishing of special values of \(L\)-functions in \(p\)-adic families.

The following references contain some more of the many applications and further results of the Ratner Theory: Einsiedler and Kleinbock (2005); Einsiedler, Margulis and Venkatesh (to appear); Elkies and McMullen (2004); Ellenberg and Venkatesh (to appear); Eskin, Masur and Schmoll (2003); Eskin and Oh (2006); Flaminio (1987); Flaminio and Spatzier (1990); Gorodnik (2004); Gorodnik and Oh (to appear); Kleinbock and Weiss (to appear); Marklof (2003); Mozes and Shah (1995); Oh (1998); Zimmer (1991).

Historical remarks

A weaker form of the Orbit Closures Theorem 2 was conjectured by M. S. Raghunathan in 1980. Theorem 1 for one-parameter \({\mathbf U}\) and lattices in reductive \({\mathbf G}\) was conjectured by S. G. Dani in 1981. Theorem <ref>th1</ref> (for lattices) and Theorem <ref>th2</ref> for general \({\mathbf G}\) and \({\mathbf U}\) were conjectured by G. A. Margulis in 1986.

It was shown earlier in Furstenberg (1972) and in Parry (1969) (see also Auslander, Green and Hahn (1963)) that Theorems 1 and 2 hold for one-parameter and one-generator subgroups of nilpotent \({\mathbf G}\). Also in Starkov (1990) Theorem 2 was proved for one-parameter \(\textrm{Ad}\)-unipotent subgroups of solvable \({\mathbf G}\) with \({\mathbf \Gamma}\) being an arbitrary closed subgroup of \({\mathbf G}\) such that \({\mathbf \Gamma}\setminus{\mathbf G}\) has finite \({\mathbf G}\)-invariant measure. Theorem 1 for the latter case follows from Starkov (1990) and Parry (1969).

As for semisimple \({\mathbf G}\), it was shown in Hedlund (1936) that if \({\mathbf G} = SL(2,{\mathbb R})\) and \({\mathbf \Gamma}\setminus{\mathbf G}\) is compact (in this case \({\mathbf \Gamma}\) is called a uniform lattice in \({\mathbf G}\)) then the action of a unipotent one-parameter subgroup \({\mathbf U}\) of \({\mathbf G}\) on \({\mathbf \Gamma}\setminus{\mathbf G}\) is minimal (i.e., every orbit of \({\mathbf U}\) is dense). Subsequently, it was proved in Furstenberg (1972) that in this case the action of \({\mathbf U}\) is uniquely ergodic.

Generalizing Furstenberg's Theorem, it was shown in Bowen (1976), Veech (1977) and Ellis and Perrizo (1978) that if \({\mathbf \Gamma}\) is a uniform lattice in a connected semisimple Lie group \({\mathbf G}\) without compact factors then ergodic actions of horospherical subgroups on \(({\mathbf \Gamma}\setminus{\mathbf G},\nu_[[:Template:\mathbf G]])\) are uniquely ergodic. Adapting the method of Furstenberg and Veech, Dani proved Theorem <ref>th1</ref> when \({\mathbf G}\) is reductive and \({\mathbf U}\) is a maximal horospherical subgroup of \({\mathbf G}\) (see Dani (1981)).

As for Theorem 2, in Dani (1986b) it was proved for horospherical subgroups of reductive \({\mathbf G}\). Also in Dani and Margulis (1990a) and Dani and Margulis (1990b) it was shown that Theorem 2 holds for one-parameter unipotent subgroups of \(SL(3,{\mathbb R})\).

Margulis proved Theorem 2 for \({\mathbf G} = SL(3,{\mathbb R})\), \({\mathbf \Gamma} = SL(3,{\mathbb Z})\), and \({\mathbf U} = SO(2,1)^0\) ( Margulis (1989)).

Theorem 3 for nilpotent \({\mathbf G}\) was proved earlier in Parry (1969) (see also Lesigne (1989)) and for \({\mathbf G} = SL(2,{\mathbb R})\) in Dani and Smillie (1984). Also in Shah (1991) it was proved for semisimple \({\mathbf G}\) of real rank \(1\) and also for some generic unipotent flows in general semisimple \({\mathbf G}\) .

It should be noted that in 1986 Dani showed (Dani (1986a)) that if \({\mathbf G}\) is a connected semisimple Lie group and \({\mathbf \Gamma}\) a lattice in \({\mathbf G}\) then given \(\epsilon > 0\) there is a compact \(K(\epsilon) \subset {\mathbf \Gamma}\setminus{\mathbf G}\) such that for any \(x \in {\mathbf \Gamma}\setminus{\mathbf G}\) and any one-parameter \(\textrm{Ad}\)-unipotent subgroup \({\mathbf U} = \{{\mathbf u}(t): t \in {\mathbb R}\}\) of \({\mathbf G}\) either \(\lambda\{t \in [0,T]: x{\mathbf u}(t) \in K(\epsilon)\} > (1-\epsilon)T\) for all large \(T\) or \(x{\mathbf L}\) is homogeneous for some proper closed connected subgroup \({\mathbf L}\) of \({\mathbf G}\) containing \({\mathbf U}\). (Here \(\lambda\) denotes the Lebesgue measure on \({\mathbb R}\).) This important result is used in the proof of Theorem 3.

Theorems 2 and 3 were published in 1991 in Ratner (1991b) and in 1993 S. G. Dani and G. A. Margulis (see Dani and Margulis (1993)) published a linearized version of the proof of Theorem 3. Using this version they showed that the convergence in Theorem 3 is uniform on compact subsets of \({\mathbf \Gamma}\setminus{\mathbf G}\).Their approach was similar to that in Shah ( 1991).

Theorems S1–S3 were first announced in 1993 in Ratner (1993) with proofs submitted at the same time and appeared in 1995 (Ratner (1995)). These proofs used the ideas and methods developed in Ratner (1990a)–Ratner (1991b).

Some of the ideas and methods from Ratner (1990a)–Ratner (1991a) were also used in Margulis and Tomanov (1994) (see also Margulis and Tomanov (1992)) to prove a particular case of Theorem S1 when each \({\mathbf G}_{\sigma}\), \(\sigma \in {\mathbb S}\) is the group of \({\mathbb Q}_{\sigma}\)-rational points of an algebraic group over \({\mathbb Q}_{\sigma}\). (The authors also formulated Theorem S3 and a weaker version of Theorem S2 for this algebraic case.) The assumption of algebraicity \({\mathbf G}_{\sigma}\) allowed them to substantially simplify the proofs. Also they considered in Margulis and Tomanov (1996) products \({\mathbf G}_[[:Template:\mathbb S]]\) of real Lie groups with finite central extensions of \(p\)-adic algebraic groups (they called the latter almost linear groups) and showed how to reduce Theorem S1 for such products to the case when \({\mathbf G}_{\sigma}\) is algebraic for every \(\sigma \in {\mathbb S}\). Since almost linear \(p\)-adic Lie groups are regular, their result (for discrete \({\mathbf \Gamma}\)) is contained in Theorem S1.

Theorems S4–S6 for horocycle flow on homogeneous spaces of \(SL(2,{\mathbb R})\) were proved in Ratner (1982a)–Ratner (1983) using the \(H\)-principle developed there (see Ratner (1984) for a survey of these results). Subsequently these theorems and the \(H\)-principle were generalized in Witte (1985)–Witte (1994) to \(\textrm{Ad}\)-unipotent flows on homogeneous spaces of arbitrary real Lie groups.

All this was done before Theorem 1 was discovered, and after it was, it became clear that Theorems S4–S5 for the real case easily follow from Theorem 1, and Theorem S6 for the real case from Theorem 1 and Witte (1994).

Likewise, Theorems S4–S6 for the general \(S\)-arithmetic setting easily follow from Theorem S1 and Witte (1994).

A simple exposition of the Ratner Theory is given in Witte–Morris (2005) (see also surveys Borel (1995); Kleinbock, Shah and Starkov (2002); Ratner (1984); Ratner (1992); Ratner (1994a); Ratner (1994b); Ratner (1998)).

References

• Auslander L., Green L. and Hahn F. (1963) Flows on homogeneous spaces, Ann. of Math. Stud., 53, Princeton Univ. Press, Princeton, NJ.

• Borel A. (1995) Values of indefinite quadratic forms at integral points and flows on spaces of lattices, Bull. Amer. Math. Soc., 32, 184–204.

• Borel A. and Prasad G. (1992) Values of isotropic quadratic forms at \(S\)-integral points, Compositio Math., 83, 347–372.

• Bowen R. (1976) Weak mixing and unique ergodicity on homogeneous spaces, Israel Math. J., 23, 267–273.

• Dani S.G. (1981) Invariant measures and minimal sets of horospherical flows, Invent. Math. , 64, 357–385.

• Dani S.G. (1986a) On orbits of unipotent flows on homogeneous spaces, II, Ergodic Theory Dynamical Systems, 6, 167–182.

• Dani S.G. (1986b) Orbits of horospherical flows, Duke Math. J., 53, 177–188.

• Dani S.G. and Margulis G.A. (1989) Values of quadratic forms at primitive integral points, Invent. Math., 98, 405–425.

• Dani S.G. and Margulis G.A. (1990a) Orbit closures of generic unipotent flows on homogeneous spaces of \(SL(3,{\mathbb R})\), Math. Ann., 286, 101–128.

• Dani S.G. and Margulis G.A. (1990b) Values of quadratic forms at integral points: An elementary approach, L'enseig. Math., 36, 143–174.

• Dani S.G. and Margulis G.A. (1993) Limit distributions of orbits of unipotent flows and values of quadratic forms, Adv. in Sov. Math., 16, 91–137.

• Dani S.G. and Smillie J. (1984) Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J., 5, 185–194.

• Einsiedler M. and Katok A. (2005) Rigidity of measures – the high entropy case and non-commuting foliations, Israel J. Math., 148, 169–238.

• Einsiedler M., Katok A. and Lindenstrauss E. (to appear) Invariant measures and the set of exceptions to Littlewood's conjecture, in Ann. of Math.

• Einsiedler M. and Kleinbock D. (2005) Measure rigidity and \(p\)-adic Littlewood-type problems, preprint (17 pages).

• Einsiedler M. and Lindenstrauss E. (2005) Joining of higher rank diagonalizable actions on locally homogeneous spaces, to appear in Duke Math. J. (29 pages).

• Einsiedler M. and Lindenstrauss E. (2006) Diagonal flows on locally homogeneous spaces and number theory, to appear in the Proceedings of the International Congress of Mathematicians 2006 (29 pages).

• Einsiedler M. and Lindenstrauss E. (2007) Diagonal actions of locally homogeneous spaces, lecture notes for CMI summer school Homogeneous Flows, Moduli Spaces and Arithmetic, in preparation.

• Einsiedler M. and Lindenstrauss E. (to appear) On measures invariant under diagonalizable actions — the rank one case and the general low entropy method.

• Einsiedler M., Margulis G.A. and Venkatesh A. (to appear) Effective equidistribution of closed orbits of semisimple groups on homogeneous spaces.

• Elkies N. and McMullen C.T. (2004) Gaps in \(\sqrt{n} \mod 1\) and ergodic theory, Duke Math. J., 123, no. 1, 95–139.

• Ellenberg J. and Venkatesh A. (to appear) Local-global principles for representations of quadratic forms.

• Ellis R. and Perrizo W. (1978) Unique ergodicity of flows on homogeneous spaces, Israel J. Math., 29, 276–284.

• Eskin A., Masur H. and Schmoll M. (2003) Billiards in rectangles with barriers, Duke Math. J., 118, no. 3, 427–463.

• Eskin A. and McMullen C. (1993) Mixing, counting and equidistribution in Lie groups, Duke Math. J., 71, 181–209.

• Eskin A., Mozes S. and Shah N. (1996) Unipotent flows and counting lattice points on homogeneous varieties, in Ann. of Math. (2), 143(2), 253–299.

• Eskin A., Margulis G.A. and Mozes S. (1998) Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147, no. 1, 93–141.

• Eskin A. and Oh H. (2006) Representation of integers by an invariant polynomial and unipotent flows, Duke Math. J., vol. 135, 481–506.

• Flaminio L. (1987) An extension of Ratner's rigidity theorem to \(n\)-dimensional hyperbolic space, Ergodic Theory Dynamical Systems, 7, 73–92.

• Flaminio L. and Spatzier R. (1990) Geometrically finite groups, Patherson–Sullivan measures and Ratner's rigidity theorem, Invent. Math., 99, 601–626.

• Furstenberg H. (1972) The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics, Lecture Notes in Math., 318, Springer–Verlag, Berlin and New York, 95–115.

• Gorodnik A. (2004) Uniform distribution of orbits of lattices on spaces of frames, Duke Math. J., 122, no. 3, 549–589.

• Gorodnik A. and Oh H. (to appear) Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary, in Duke Math. J.

• Hedlund G.A. (1936) Fuchsian groups and transitive horocycles, Duke Math. J., 2, 530–542.

• Kleinbock D., Shah N. and Starkov A. (2002) Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, in Handbook of Dynamical Systems, vol. 1A, 813–930, North Holland, Amsterdam.

• Kleinbock D. and Weiss B. (to appear) Dirichlet's Theorem on diophantine approximation and homogeneous flows.

• Lesigne E. (1989) Th\'eor\`emes ergodiques pour une translation sur nilvari\'et\'e, Ergodic Theory Dynamical Systems, 9, 115–126.

• Lindenstrauss E. (2006) Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2), 163, 165–219.

• Marklof J. (2003) Pair correlation densities of inhomogeneous quadratic forms, Ann. of Math., 158, no. 2, 419–471.

• Margulis G.A. (1989) Discrete subgroups and ergodic theory, in Number Theory, Trace Formula and Discrete Groups, Symposium in honor of A. Selberg, Oslo 1987, Academic Press, New York, 377–398.

• Margulis G.A. (1990) Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, Proc. Internat. Congress Math., Kyoto, 193–215.

• Margulis G.A. and Tomanov G.M. (1992) Measure rigidity for algebraic groups over local fields, C.R. Acad. Sci. Paris, t. 315, S\'erie I, 1221–1226.

• Margulis G.A. and Tomanov G.M. (1994) Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116, 347–392.

• Margulis G.A. and Tomanov G.M. (1996) Measure rigidity for almost linear groups and its applications, Journal d'Analyse Math., 69, 25–54.

• Mozes S. (1995) Epimorphic subgroups and invariant measures, Ergodic Theory Dynamical Systems, 15, 1–4.

• Mozes S. and Shah N. (1995) On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynamical Systems, 15, 149–159.

• Oh H. (1998) Discrete subgroups generated by lattices in opposite horospherical subgroups, J. Algebra, 203, no. 2, 621–676.

• Parry W. (1969) Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math., 91, 757–771.

• Parry W. (1971) Metric classifications of ergodic nilflows and unipotent affines, Amer. J. Math., 93, 819–828.

• Ratner M. (1982a) Rigidity of horocycle flows, Ann. of Math. (2), 115, 597–614.

• Ratner M. (1982b) Factors of horocycle flows, Ergodic Theory Dynamical Systems, 2, 465–489.

• Ratner M. (1983) Horocycle flows: Joinings and rigidity of products, Ann. of Math. (2), 118, 277–313.

• Ratner M. (1984) Ergodic theory in hyperbolic space, Contemp. Math., 26, 302–334.

• Ratner M. (1990a) Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math., 101, 449–482.

• Ratner M. (1990b) On measure rigidity of unipotent subgroups of semisimple groups, Acta Math., 165, 229–309.

• Ratner M. (1991a) On Raghunathan's measure conjecture, Ann. of Math. (2),134, 545–607.

• Ratner M. (1991b) Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63, 235–280.

• Ratner M. (1992) Raghunathan's conjectures for \(SL(2,{\mathbb R})\), Israel J. Math., 80, 1–31.

• Ratner M. (1993) Raghunathan's conjectures for \(p\)-adic Lie groups, International Mathematics Research Notices, no. 5, 141–146.

• Ratner M. (1994a) Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Functional Anal. (GAFA), 4, 236–256.

• Ratner M. (1994b) "Interactions Between Ergodic Theory, Lie Groups and Number Theory," Proceedings of ICM 1994, Z\"urich.

• Ratner M. (1995) Raghunathan's conjectures for Cartesian products of real and \(p\)-adic Lie groups, Duke Math. J., 77, 275–382.

• Ratner M. (1998) On the \(p\)-adic and \(S\)-arithmetic generalizations of Raghunathan's conjectures, Proc. of Intern. Colloquium on Lie Groups and Ergodic Theory, Tata Institute of Fundamental Research, Mumbay, India.

• Shah N. (1991) Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., 289, 315–334.

• Shah N. (1994) Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J., 75, 711–732.

• Shah N. and Weiss B. (2000) On actions of epimorphic subgroups on homogeneous spaces, Ergodic Theory and Dynamical Systems, 20, no. 2, 567–592.

• Starkov A. (1989) Solvable homogeneous flows, Math. USSR-Sb., 62, 243–260.

• Starkov A. (1990) Structure of orbits of homogeneous flows and Raghunathan's conjecture, Russian Math: Surveys, 45, 227–228.

• Vatsal V. (2002) Uniform distribution of Heegner points, Invent. Math., 148, 1–46.

• Vatsal V. (2003) Special values of anticylotomic \(L\)-functions, Duke Math. J. (2), 116, 219–261.

• Vatsal V. (2004) Special value formulae for Rankin \(L\)-functions, in Heegner Points and Rankin \(L\)-Series, volume 49 of Math. Sci. Res. Inst. Publ., pages 165–190, Cambridge Univ. Press, Cambridge.

• Vatsal V. (2006) Special values of \(L\)-functions modulo \(p\), to appear in Proc. of ICM.

• Veech W. (1997) Unique ergodicity of horospherical flows, Amer. J. Math., 99, 827–859.

• Witte D. (1985) Rigidity of some translations on homogeneous spaces, Invent. Math., 81, 1–27.

• Witte D. (1987) Zero entropy affine maps on homogeneous spaces, Amer. J. Math., 109, 927–961.

• Witte D. (1989) Rigidity of horospherical foliations, Ergodic Theory Dynamical Systems, 9, 191–205.

• Witte D. (1994) Measurable quotients of unipotent translations on homogeneous spaces, Trans. Amer. Math. Soc., 345, 577–594.

• Witte–Morris D. (2005) Ratner's Theorems on unipotent flows, Chicago Lectures in Mathematics.

• Zimmer R.J. (1991) Superrigidity, Ratner's theorem, and fundamental groups, Israel J. Math., 74, no. 2–3, 199–207.

See also

Ergodic Theory, Dynamical Systems