In this blog, we mark some article that is important in the investigation of the equidistribution problem, each of them providing new inspirit to solve some particuler type of equidistribution problem.
1. Mixing, counting, and equidistribution in Lie groups
The author is Alex Eskin, Curt McMullen
This article was published in 1993. in Duke Math. J. 71(1): 181-209 (July 1993). DOI: 10.1215/S0012-7094-93-07108-6
Let $\Gamma \subset G=\operatorname{Aut}\left(\mathbb{H}^2\right)$ be a group of isometries of the hyperbolic plane $\mathbb{H}^2$ such that $\Sigma=\Gamma \backslash \mathbb{H}^2$ is a surface of finite area. Then:
I. The geodesic flow is mixing on the unit tangent bundle $T_1(\Sigma)=\Gamma \backslash G$.
II. The sphere $S(x, R)$ of radius $R$ about a point $x \in \Sigma$ becomes equidistributed as $R \rightarrow \infty$.
III. The number of points $N(R)$ in an orbit $\Gamma v$ which lie within a hyperbolic ball $B(p, R) \subset \mathbb{H}^2$ has the asymptotic behavior
$$
N(R) \sim \frac{\operatorname{area}(B(p, R))}{\operatorname{area}(\Sigma)} .
$$
This is a classical result, and this article establishes a similar theorem in local symmetric space.
The purpose of the article, Mixing, counting, and equidistribution in Lie groups, is to discuss results similar to those above where the hyperbolic plane is replaced by a general affine symmetric space $V=G / H$. This setting includes the classical Riemannian symmetric spaces (when $H$ is a maximal compact subgroup) as well as spaces with indefinite invariant metrics.
A simple non-Riemannian example is obtained by letting $V$ be the space of oriented geodesics in the hyperbolic plane. Then $H=A$, the group of diagonal matrices in $G=P S L_2(\mathbb{R})$. In this case $\Gamma \backslash G / H$ is not even Hausdorff.
This setting includes counting theorems for integral points on a large class of homogeneous varieties (e.g. those associated to quadratic forms) and allows us to prove some of the main theorems of [DRS] by elementary arguments (see $\$ 6$ ).
Statement of Results. Let $G$ be a connected semisimple Lie group with finite center and let $H \subset G$ be a closed subgroup such that $G / H$ is an affine symmetric space (cf. $[\mathrm{F}-\mathrm{J}]$, [Sch]). This means there is an involution $\sigma: G \rightarrow G$ such that $H$ is the fixed-point set of $\sigma$ :
$$
H={g: \sigma(g)=g} .
$$
(By involution we mean a Lie group automorphism such that $\sigma^2=\mathrm{id}$.)
Let $\Gamma \subset G$ be a lattice, i.e. a discrete subgroup such that the volume of $X=\Gamma \backslash G$ is finite.
Statement of Results. Let $G$ be a connected semisimple Lie group with finite center and let $H \subset G$ be a closed subgroup such that $G / H$ is an affine symmetric space (cf. $[\mathrm{F}-\mathrm{J}],[\mathrm{Sch}]$ ). This means there is an involution $\sigma: G \rightarrow G$ such that $H$ is the fixed-point set of $\sigma$ :
$$
H={g: \sigma(g)=g} .
$$
(By involution we mean a Lie group automorphism such that $\sigma^2=\mathrm{id}$.) Let $\Gamma \subset G$ be a lattice, i.e. a discrete subgroup such that the volume of $X=\Gamma \backslash G$ is finite.
Assume further that $\Gamma$ has dense projection to $G / G^{\prime}$ for any positive-dimensional normal noncompact Lie subgroup $G^{\prime} \subset G$. $^1$
Finally, assume that $\Gamma$ meets $H$ in a lattice: that is, the volume of $Y=(\Gamma \cap H) \backslash H$ is finite.
We may now state general results on mixing, equidistribution, and counting. The mixing theorem below is standard; the aim of this paper is to deduce the equidistribution and counting theorems from it, using the geometry of affine symmetric spaces.
THEOREM $1.1$ (Mixing). The action of $G$ on $X=\Gamma \backslash G$ is mixing. That is, for any $\alpha, \beta$ in $L^2(X)$
$$
\int_X \alpha(x g) \beta(x) d x \rightarrow \frac{\int_X \alpha \int_X \beta}{m(X)}
$$
as $g$ tends to infinity.
Here the integrals and the volume $m(X)$ are taken with respect to the $G$-invariant Haar measure on $X$. A sequence of elements $g_n \in G$ tends to infinity if any compact set $K \subset G$ contains only finitely many terms in the sequence.
THEOREM $1.2$ (Equidistribution). The translates $Y g$ of the $H$-orbit
$$
Y=(\Gamma \cap H) \backslash H
$$
become equidistributed on $X=\Gamma \backslash G$ as $g$ tends to infinity in $H \backslash G$. That is,
$$
\frac{1}{m(Y)} \int_{Y_g} f(h) d h \rightarrow \frac{1}{m(X)} \int_X f(x) d x
$$
as $g$ leaves compact subsets of $H \backslash G$.
To state the theorem on counting points in an orbit, we first isolate some properties of the sets used for counting. Let $B_n \subset G / H$ be a sequence of finite volume measurable sets such that the volume of $B_n$ tends to infinity.
Definition. The sequence $B_n$ is well rounded if for any $\varepsilon>0$ there exists an open neighborhood $U$ of the identity in $G$ such that
$$
\frac{m\left(U \cdot \partial B_n\right)}{m\left(B_n\right)}<\varepsilon $$ for all $n$. It is easy to verify the following statement. Proposition 1.3. A sequence is well rounded if and only if for any $\varepsilon>0$ there is a neighborhood $U$ of $\mathrm{id} \in G$ such that, for all $n$,
$$
(1-\varepsilon) m\left(\bigcup_U g B_n\right)<m\left(B_n\right)<(1+\varepsilon) m\left(\bigcap_U g B_n\right) \text {. }
$$
That is, $B_n$ is nearly invariant under the action of a small neighborhood of the identity.
THEOREM $1.4$ (Counting). Let $V=G / H$ be an affine symmetric space and let $v$ denote the coset $[\mathrm{H}]$. For any well-rounded sequence, the cardinality of the number of points of $\Gamma v$ which lie in $B_n$ grows like the volume of $B_n$ : asymptotically,
$$
\left|\Gamma v \cap B_n\right| \sim \frac{m((\Gamma \cap H) \backslash H)}{m(\Gamma \backslash G)} m\left(B_n\right)
$$