Pointwise Equidistribution and Translates of Measures on Homogeneous Spaces

TL;DR

This paper proves that under broad conditions, the empirical measures of orbits under a sequence of transformations on homogeneous spaces become equidistributed towards a given measure, with applications to sparse horocycle flows.

Contribution

It establishes pointwise equidistribution results for translates of measures on homogeneous spaces, extending previous work to sequences with full upper density and applying to sparse orbit distributions.

Findings

Empirical measures of orbits converge to the invariant measure under general conditions.

Results apply to translates of closed orbits of Lie groups on homogeneous spaces.

Proves equidistribution of exponentially sparse horocycle orbits starting from almost every point.

Abstract

Let $(X,\mathfrak{B},\mu)$ be a Borel probability space. Let $T_n: X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $\nu$ be a probability measure on $X$ such that $\frac{1}{N}\sum_{n=1}^N (T_n)_\ast \nu \rightarrow \mu$ in the weak-$\ast$ topology. Under general conditions, we show that for $\nu$ almost every $x\in X$, the measures $\frac{1}{N}\sum_{n=1}^N \delta_{T_n x}$ get equidistributed towards $\mu$ if $N$ is restricted to a set of full upper density. We present applications of these results to translates of closed orbits of Lie groups on homogeneous spaces. As a corollary, we prove equidistribution of exponentially sparse orbits of the horocycle flow on quotients of $SL(2,\mathbb{R})$, starting from every point in almost every direction.

Equations

$$\frac{1}{N}\sum _{n=1}^N(T_n{)}_{\ast }\nu \stackrel{N\to \infty }{\to }\mu$$

$$\frac{1}{N}\sum _{n=1}^N{\delta }_{T_{n}x}$$

$$\overline{d}(A)=\underset{N\to \infty }{\mathrm{limsup}}\frac{\#(A\cap [1,N])}{N}$$

$$\frac{1}{N}\sum _{n=0}^{N-1}{T}_{\ast }^n\nu \underset{N\to \infty }{\overset{weak-\ast }{\to }}\mu$$

$$\underset{\begin{array}{c}N\to \infty \\ N\in A(x)\end{array}}{\lim }\frac{1}{N}\sum _{n=0}^{N-1}\psi (T^{n}x)=\int \psi d\mu$$

$$\underset{\begin{array}{c}N\to \infty \\ N\in A(x)\end{array}}{\lim }\frac{1}{N}\sum _{n=1}^N\psi (T_{n}x)=\int \psi d\mu$$

$$\frac{1}{N}\sum _{n=1}^N(g_n{)}_{\ast }{\mu }_H\stackrel{N\to \infty }{\to }{\mu }_{G/\Gamma }$$

$$\underset{\begin{array}{c}N\to \infty \\ N\in A(x)\end{array}}{\lim }\frac{1}{N}\sum _{n=1}^N{\delta }_{g_{n}x}={\mu }_{G/\Gamma }$$

$$t_n=O\left(e^{\lambda n}\right)$$

$$g_n=\left(\begin{array}{cc}1& t_n\\ 0& 1\end{array}\right),k_{\theta }=\left(\begin{array}{cc}\cos (\theta )& \sin (\theta )\\ -\sin (\theta )& \cos (\theta )\end{array}\right)$$

$$\underset{\begin{array}{c}N\to \infty \\ N\in A(\theta )\end{array}}{\lim }\frac{1}{N}\sum _{n=1}^N{\delta }_{g_n{k}_{\theta }x}={\mu }_{G/\Gamma }$$

$$\frac{1}{N}\sum _{n=0}^{N-1}(T_n{)}_{\ast }\nu \underset{N\to \infty }{\overset{weak-\ast }{\to }}\mu$$

$$E_{\alpha ,N,j}^{\psi }=\left\{x\in X:\underset{1\le M\le N}{\sup }\left|\frac{1}{M}\sum _{k=j}^{j+M-1}\psi (T_{k}x)\right|>\alpha \right\}$$

$$\alpha \beta \nu (E_{\alpha ,N,j_N}^{\psi })\le 12||\psi |{|}_{L^1(\mu )}$$

$${\varphi }^{\ast }(a)=\underset{N\ge 1}{\sup }\left|\frac{1}{N}\sum _{i=0}^{N-1}\varphi (i+a)\right|$$

$$E_{\alpha }=\left\{a\in \mathbb{Z}\mid {\varphi }^{\ast }(a)>\alpha \right\}$$

$$\alpha \left|E_{\alpha }\right|\le 3||\varphi |{|}_{l^1(\mathbb{Z})}$$

$$\varphi (j)=\{\begin{array}{ll}\psi (T_{j}x)& 0\le j\le J\\ 0& otherwise\end{array}$$

$${\varphi }^{\ast }(a)=\underset{1\le M}{\sup }\left|\frac{1}{M}\sum _{k=0}^{M-1}\varphi (k+a)\right|,$$

$$E_{\alpha }^{\varphi }=\left\{a\in \mathbb{Z}\mid {\varphi }^{\ast }(a)>\alpha \right\},$$

$$\alpha \left|E_{\alpha ,N}^{\varphi }\right|\le \alpha \left|E_{\alpha }^{\varphi }\right|\le 3||\varphi |{|}_{l^1(\mathbb{Z})}$$

$${\varphi }_N^{\ast }(a)=\underset{1\le n\le N}{\sup }\left|\frac{1}{n}\sum _{k=0}^{n-1}\psi (T_{k+a}x)\right|$$

$${\chi }_{\alpha ,N,j}(x)=1\text{ if and only if }j\in E_{\alpha ,N}^{\varphi }$$

$$\alpha \sum _{j=0}^{J-N-1}{\chi }_{\alpha ,N,j}(x)$$

$$\alpha \sum _{j=0}^{J-N-1}\nu (E_{\alpha ,N,j}^{\psi })\le 3\sum _{j=0}^J\int |\psi (T_{j}x)|d\nu (x)$$

$$\frac{\alpha }{\beta N}\sum _{j=0}^{\beta N-1}\nu (E_{\alpha ,N,j}^{\psi })$$

$$\le \frac{6}{\beta }\frac{1}{(1+\beta )N+1}\sum _{j=0}^{(1+\beta )N}\int |\psi (T_{j}x)|d\nu (x)$$

$$\frac{1}{(1+\beta )N+1}\sum _{j=0}^{(1+\beta )N}\int |\psi (T_{j}x)|d\nu (x)\to \int |\psi |d\mu =||\psi |{|}_{L^1(\mu )}$$

$$\frac{1}{(1+\beta )N+1}\sum _{j=0}^{(1+\beta )N}\int |\psi (T_{j}x)|d\nu (x)\le 2||\psi |{|}_{L^1(\mu )}$$

$$\frac{1}{\beta N}\sum _{j=0}^{\beta N-1}\nu (E_{\alpha ,N,j}^{\psi })$$

$$\underset{n\to \infty }{\mathrm{limsup}}{A}_n=\bigcap _{n\ge 1}\bigcup _{k\ge n}{A}_k$$

Full text

1. Introduction

Many problems in number theory and geometry can be recast in terms of the equidistribution of translates of appropriate measures on quotients of certain Lie groups. The general set up of these results is a Borel probability space $(X,\mathfrak{B},\mu)$, a probability measure $\nu$ on $X$ (usually singular with respect to $\mu$) and a sequence of transformations $T_{n}:X\rightarrow X$ such that

[TABLE]

where $(T_{n})_{\ast}\nu$ is the pushforward of $\nu$ under $T_{n}$ and the convergence is in the weak-$\ast$ topology. A natural question is to what extent can one extend such results to describe the behavior of measures of the form

[TABLE]

for $\nu$-almost every $x$, where $\delta_{y}$ denotes the dirac delta measure at a point $y$.

Recently, an analogous question for flows was addressed by Chaika and Eskin [CE] in the context of flat surfaces. In that setting, $X$ is some affine submanifold of the moduli space of flat structures on a surface, $\mu$ is a natural affine $SL(2,\mathbb{R})$ invariant measure, $\nu$ is the measure supported on an orbit of $SO(2)$ and the transformations are of the form $a(t)=diag(e^{t},e^{-t})$. They show that for $\nu$ almost every $x$, the empirical measures analogous to those in (1.2) get equidistributed towards $\mu$.

In the context of homogeneous spaces, Shi [Shi] explored this question for translates of measures supported on (pieces of) orbits of certain horospherical subgroups of Lie groups by one parameter diagonalizable subgroups. Here $X$ is a homogeneous space for a Lie group $G$, $\nu$ is a measure on an orbit of a certain horospherical subgroup, and $T_{n}=T^{n}$, where $T$ is an Ad-diagonalizable element of $G$. Equidistribution of empirical measures towards the natural $G$-invariant Haar measure is proven $\nu$ almost everywhere.

In [KSW], an effective version of this result is obtained via different methods. The convergence of empirical measures analogous to (1.2) is proven for general dynamical systems under the hypothesis of some form of exponential mixing of the transformation $T$ with respect to the non-invariant measure $\nu$.

In all three cases, equidistribution was obtained by exploiting specific properties of the system at hand, while not directly utilizing the fact that (1.1) holds.

1.1. Statement of Results

In this article, we approach this question in the general context of continuous measure preserving transformations, assuming (1.1) only. We obtain equidistribution results of measures in (1.2) under general conditions yet only along subsequences of full upper density. Recall the definition of upper density:

Definition**.**

The upper density of a subset $A\subseteq\mathbb{N}$, denoted by $\overline{d}(A)$ is defined to be

[TABLE]

In what follows, $X$ will be a locally compact, second countable topological space and $\mathcal{B}$ is its Borel $\sigma$-algebra. A pair $(X,\mathcal{B})$ will be called a standard Borel space. The following is our first main result for the case when translations are done by powers of a single transformation.

Theorem 1.1**.**

Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $T$ be a continuous ergodic measure preserving transformation of $X$. Let $\nu$ be a probability measure on $X$ such that

[TABLE]

Then, for $\nu$-almost every $x\in X$, there exists a sequence $A(x)\subseteq\mathbb{N}$, of upper density $1$, such that for all $\psi\in C_{c}(X)$,

[TABLE]

The proof of Theorem 1.1 relies on an adaptation of the weak-type maximal inequality and follows similar lines to the proof of the classical Birkhoff ergodic theorem.

Our next result concerns the more general situation of translating by sequences of transformations. In this generality, we assume more structure on the possible limit points of the empirical measures. It would be interesting to know if a similar statement is still valid under weaker, more general hypotheses.

Theorem 1.2**.**

Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $(T_{n})_{n}$ be a sequence of continuous transformations of $X$. Let $S:X\rightarrow X$ be a continuous ergodic $\mu$ preserving transformation. Let $\nu$ be a probability measure on $X$. Assume the following holds:

1. (1)

$\frac{1}{N}\sum_{n=1}^{N}(T_{n})_{\ast}\nu\xrightarrow{N\rightarrow\infty}\mu$* in the weak-$\ast$ topology.* 2. (2)

For $\nu$-almost every $x\in X$, any limit point of the sequence of measures $\frac{1}{N}\sum_{n=1}^{N}\delta_{T_{n}x}$ is $S$-invariant. 3. (3)

There exists a Borel measurable, $\sigma$-compact set $Z\in\mathcal{B}$ such that $\mu(Z)=0$ and for all ergodic $S$ invariant measures $\lambda\neq\mu$, $\lambda(Z)=1$.

Then, for $\nu$-almost every $x\in X$, there exists a sequence $A(x)\subseteq\mathbb{N}$, of upper density equal to $1$, such that for all $\psi\in C_{c}(X)$,

[TABLE]

We remark that the maximal inequality used in the proof of Theorem 1.2 does not require the hypotheses of the Theorem.

We now discuss some applications of these results. Theorem 1.1 is broadly applicable to general dynamical systems and thus we present some applications of Theorem 1.2 within homogeneous dynamics where we demonstrate that its hypotheses are verified.

1.2. Sparse Equidistribution and Translates of Orbits of Lie Groups

Our motivation for studying this question comes from the problem of sparse equidistribution of unipotent flows on homogeneous spaces. This was conjectured by Shah in [Sha]. Recent progress was achieved in [Ven] for the horocycle flow on compact quotients of $SL_{2}(\mathbb{R})$ along sequences of the form $n^{1+\gamma}$ for small values of $\gamma$. See also the work of Sarnak and Ubis on the equidistribution along the primes [SU] on $SL_{2}(\mathbb{R})/SL_{2}(\mathbb{Z})$. However, the question in full generality remains open.

With the help of Theorem 1.2, we obtain an equidistribution result for exponentially sparse orbits of unipotent flows, of which the horocycle flow is an example. We will need some notation to state our results precisely.

Let $G$ be a connected semisimple Lie group and let $\Gamma$ be a lattice in $G$. Let $\mu_{G/\Gamma}$ denote the unique $G$ invariant Haar probability measure on $G/\Gamma$. Let $H$ be a closed connected subgroup such that the orbit $H\Gamma$ is closed in $G/\Gamma$ and supports a probability $H$-invariant measure, denoted by $\mu_{H}$.

In order to apply Theorem 1.2 in this setup, we introduce the notion of Ratner Sequences. We say a sequence $g_{n}$ of elements of $G$ is a Ratner Sequence with respect to $H$ if there exists a non-trivial $Ad$-unipotent element $u\in G$ such that for $\mu_{H}$ almost every $x\in G/\Gamma$, any limit point of the sequence of empirical measures $\frac{1}{N}\sum_{n=1}^{N}\delta_{g_{n}x}$ is invariant by $u$.

Our main theorem in this set up is the following.

Theorem 1.3**.**

Let $g_{n}$ be a Ratner Sequence with respect to $H$ satisfying

[TABLE]

Then, for $\mu_{H}$ almost every $x\in G/\Gamma$, there exists a sequence $A(x)\subseteq\mathbb{N}$ of upper density $1$ such that

[TABLE]

Apart from Theorem 1.2, a key ingredient in the proof of Theorem 1.3 is Ratner’s measure classification theorem [Rat].

The assumption on the equidistribution of the translates of $\mu_{H}$ by a sequence of elements $g_{n}$ is satisfied in numerous examples in homogeneous dynamics. For example, it was shown in [EM] that (1.3) is satisfied as soon as $H$ is a symmetric subgroup of $G$ and $g_{n}$ tends to infinity in $G/H$. Recall that $H$ is said to be a symmetric subgroup if it is the fixed point set of an involution of $G$.

The result of Eskin and McMullen was significantly strengthened in [EMS] to include translates of reductive subgroups that is, subgroups which are only invariant by an involution of $G$. Note that we only require $H$ to be connected in Theorem 1.3. Thus, the applicability of Theorem 1.3 boils down to the existence of Ratner sequences which is the subject of the next section.

1.2.1. Existence of Ratner Sequences

We prove a general criterion on the existence of sparse Ratner Sequences inside $1$-parameter unipotent subgroups. The proof relies on the representation theory of embedded copies of $SL_{2}(\mathbb{R})$ inside $G$ in addition to an adaptation of a technique developed by Chaika and Eskin [CE] in the setting of strata of Abelian differentials.

Let the notation be the same as above and let $Z_{G}(H)$ denote the centralizer of $H$ inside $G$. The following is the main statement.

Theorem 1.4**.**

Let $U=\left\{u_{t}:t\in\mathbb{R}\right\}$ be a $1$-parameter $Ad$-unipotent subgroup of $G$ such that $U\not\subset Z_{G}(H)$. Suppose $t_{n}>0$ is a sequence satisfying

[TABLE]

for some constant $\lambda>0$ and all $n\geq 1$, Then, $u_{t_{n}}$ is a Ratner sequence for $H$.

The exponential growth condition is used in the adaptation of Chaika-Eskin’s technique which is similar in spirit to Breiman’s law of large numbers. It uses the law of large numbers for weakly dependent random variables. Exponential growth guarantees a sufficiently fast rate of decay of correlations.

In the appendix, we prove a more crude criterion for the existence of Ratner sequences in the setting where $H$ is a symmetric subgroup of $G$, but we drop the restriction on the unipotence of the elements $g_{n}$.

We show that any sequence $g_{n}$ satisfying an exponential growth condition similar to the one in Theorem 1.4 contains a Ratner sequence as a subsequence. See Theorem A.1 for the precise statement.

In the particular instance when $G=SL_{2}(\mathbb{R})$ and the sequence $g_{n}$ comes from the action of the horocycle flow, Theorem 1.3 takes the following form:

Corollary 1.5**.**

Let $G=SL(2,\mathbb{R})$, $\Gamma\subset G$ a lattice and let $K=SO(2)$. Let $\lambda>0$ and for $n\in\mathbb{N}$, let $t_{n}=e^{\lambda n}$. Let

[TABLE]

Then, for every $x\in G/\Gamma$ and for almost every $\theta\in[0,2\pi]$, there exists a sequence $A(\theta)\subseteq\mathbb{N}$ of upper density $1$ such that

[TABLE]

Moreover, if $G/\Gamma$ is compact then $A(\theta)=\mathbb{N}$.

Proof.

Let $H=SO(2)$. Since $H$ is its own centralizer in $G$, by Theorem 1.4, $g_{n}$ is a Ratner sequence for $H$. By the work of Eskin and McMullen [EM], one has that $g_{n}\mu_{H}\rightarrow\mu_{G/\Gamma}$. Thus, the corollary follows from Theorem 1.3. When $G/\Gamma$ is compact, the action by unipotent elements on $G/\Gamma$ is uniquely ergodic in this case, the Haar measure being the only invariant measure. Moreover, the space of probability measures on $G/\Gamma$ is weak-$\ast$ compact and hence all limit points of the sequences of empirical measures are probability measures. Since $g_{n}$ is a Ratner sequence, the claim follows. ∎

The main point is that this holds for every $x$. This is not guaranteed by any general theorem on sparse equidistribution almost everywhere.

The paper is organized as follows. In § 2, we prove an analogue of the maximal ergodic theorem in our set up. We use this to prove Theorems 1.1 and 1.2 in § 3 and § 4. In § 5, we discuss the existence of Ratner sequences and prove Theorem 1.4. In § 6, we provide a proof of Theorem 1.3.

Acknowledgement**.**

I would like to thank my advisor, Nimish Shah, for numerous valuable discussions.

2. An Analogue of the Maximal Inequality

The following proposition is an extension of the classical maximal ergodic theorem to the set up involving sequences of transformations and a non-invariant measure.

Proposition 2.1**.**

Let $(X,\mathcal{B})$ be a standard Borel probability space. Let $T_{n}$ be a sequence of continuous transformations on $X$. Let $\nu,\mu$ be probability measures on $X$ such that

[TABLE]

Let $\psi\in C_{c}(X)$ and let $\alpha>0$ and $\beta\in(0,1)$. For every $j,N\geq 1$, define the set

[TABLE]

Then, for every $N\gg 1$, depending on $\psi$, there exists some $0\leq j_{N}<\beta N$, such that

[TABLE]

We will deduce this proposition from the classical maximal inequality for $l^{1}(\mathbb{Z})$ which is a consequence of Vitali’s covering lemma. The proof follows closely the proof of the usual maximal inequality.

Lemma 2.2** (Lemma 2.29, [EW]).**

Let $\phi\in l^{1}(\mathbb{Z})$. Define the following maximal function, for $a\in\mathbb{Z}$:

[TABLE]

Let $\alpha>0$ and define

[TABLE]

Then,

[TABLE]

Proof of Proposition 2.1.

Let $\psi\in C_{c}(X)$ and let $\alpha>0$, $\beta\in(0,1)$. Let $N\geq 1$ and let $E_{\alpha,N,j}$ be as in the statement. Let $x\in X$ and let $J>N$ be a parameter to be determined later. Define the following function

[TABLE]

Then, clearly $\phi\in l^{1}(\mathbb{Z})$. For $a\in\mathbb{Z}$, define the following two functions

[TABLE]

Define the corresponding exceptional sets

[TABLE]

By Lemma 2.2 applied to $\phi$, we have

[TABLE]

Note that for $a\in[0,J-N-1]$, we have

[TABLE]

Let $\chi_{{\alpha,N,j}}$ denote the indicator function of $E_{\alpha,N,j}^{\psi}$. Thus, for $j\in[0,J-N-1]$,

[TABLE]

Thus, combining (2.1), (2.2) and (2.3) along with the definition of $\phi$, we get

[TABLE]

Integrating both sides of the above with respect to $\nu$ yields

[TABLE]

Taking $J=(1+\beta)N$ in (2.4) and dividing both sides by $J-N$

[TABLE]

Now, by assumption,

[TABLE]

Thus, for all $N$ sufficiently large, depending only on $\psi$, we have

[TABLE]

Combining this with (2), we get for all $N$ sufficiently large,

[TABLE]

Thus, there must exist some $j=j(N)\in[0,\beta N-1]$ for which the conclusion of the Proposition holds.

∎

3. An Analogue of Birkhoff’s Ergodic Theorem

This section is dedicated to the proof of Theorem 1.1. With the maximal inequality for the non-invariant measure $\nu$ in place (Proposition 2.1), our proof will follow the same lines as Bourgain’s approach to deduce the classical Birkhoff theorem from the mean ergodic theorem (cf. [Bou], Section 2-C).

Recall that for a sequence of sets $A_{n}$, the $limsup$ of these sets is the set of elements which belong to $A_{n}$ for infinitely many $n$. More precisely,

[TABLE]

The following simple observation will be used repeatedly in what follows.

Lemma 3.1**.**

Let $X$ be a standard Borel space and let $\mu$ be a probability measure on $X$. Let $A_{n}\subseteq X$ be a sequence of measurable sets such that $\mu(A_{n})\geq\alpha$ for some $\alpha\in[0,1]$. Then,

[TABLE]

Proof.

This follows from the definition of $\limsup_{n\rightarrow\infty}A_{n}$ as a decreasing intersection and the continuity of the measure $\mu$. ∎

The following Lemma is the main step in the proof of Theorem 1.1.

Lemma 3.2**.**

Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $T$ be an ergodic measure preserving transformation on $X$. Let $\nu$ be a probability measure on $X$ such that

[TABLE]

Let $f_{1},\dots,f_{n}\in C_{c}(X)$. Then, for $\nu$-almost every $x\in X$, there exists a sequence $A\subseteq\mathbb{N}$, of upper density $1$, depending on $x$ and the functions $f_{1},\dots,f_{n}$, such that for all $k=1,\dots,n$,

[TABLE]

Let us deduce Theorem 1.1 from this Lemma first.

3.1. Proof of Theorem 1.1

Let $\mathcal{F}=\left\{f_{k}\in C_{c}(X):k\in\mathbb{N}\right\}$ be an enumeration of a countable set of continuous functions which are dense in $C_{c}(X)$ in the uniform norm. Then, it suffices to show that for $\nu$ almost every $x\in X$, there exists a sequence $A(x)\subseteq\mathbb{N}$, of full upper density such that for all $f\in\mathcal{F}$,

[TABLE]

For each $n\in\mathbb{N}$, let $\mathcal{F}_{n}=\left\{f_{1},\dots,f_{n}\right\}\subset\mathcal{F}$. By Lemma 3.2, for each $n$, there exists a set $X_{n}$ with $\nu(X_{n})=1$, such that for all $x\in X_{n}$, there exists a sequence $A(x,\mathcal{F}_{n})\subseteq\mathbb{N}$, along which the limit in (3.1) holds for all $f\in\mathcal{F}_{n}$.

Let $Y=\cap_{n}X_{n}$. Then, $\nu(Y)=1$. Let $y\in Y$. We will build a sequence $A(y)$ by induction from the sequences $A(y,\mathcal{F}_{n})$ via a standard argument which we will make use of several times later.

For each $n\in\mathbb{N}$, let $N_{n}\in\mathbb{N}$ be such that for all $N\geq N_{n}$ with $N\in A(y,\mathcal{F}_{n})$, and all $f\in\mathcal{F}_{n}$,

[TABLE]

Let $M_{1}=N_{1}$. If $M_{j}$ has been defined, let $M_{j+1}$ be such that the following holds

[TABLE]

Note that the above implies that

[TABLE]

Now, define the sequence $A(y)$ as follows:

[TABLE]

Thus, by construction, the upper density of $A(y)$ is equal to $1$. Now, let $f\in\mathcal{F}$. Then, $f\in\mathcal{F}_{n}$ for all $n\geq n_{0}$, for some $n_{0}\in\mathbb{N}$. Thus, for all $N\geq M_{n_{0}}$ such that $N\in A(y)$, there exists $j\geq n_{0}$, such that $N\in A(y,\mathcal{F}_{j})\cap[M_{j},M_{j+1}]$. Thus, since $M_{j}\geq N_{j}$, by (3.2), the conclusion follows.

3.2. Proof of Lemma 3.2

For any function $\psi$ and for every $N\geq 1$, let $\mu(\psi)=\int\psi\;d\mu$ and let

[TABLE]

Let $\varepsilon\in(0,1)$. By Von Neumann’s mean ergodic theorem, for all $k$,

[TABLE]

Hence, we can find some $M\gg 1$, for all $k\leq n$,

[TABLE]

Let $\beta=\frac{\varepsilon}{C}$, where

[TABLE]

Let $g_{k}=A_{M}(f_{k})-\mu(f_{k})$. Note that $||g_{k}||_{\infty}\leq C$. For all $k$ and for every $N\in\mathbb{N}$, define

[TABLE]

Then, by the analogue of the maximal inequality, Proposition 2.1, applied to $g_{k}$, the sequence of transformations $T_{l}=T^{l}$ and $E_{\varepsilon,N,j}=T^{-j}E_{\varepsilon,N}^{k}$, for all $N$ sufficiently large, depending on $\varepsilon$, there exists $j_{N,k}\in[0,\beta N]$ such that

[TABLE]

Let $G_{N,k}^{\varepsilon}=X\setminus T^{-j_{N,k}}E_{\varepsilon,N}^{k}$ and let $G_{N}^{\varepsilon}=\bigcap_{k=1}^{n}G_{N,k}^{\varepsilon}$. Thus, by (3.3) and Lemma 3.1,

[TABLE]

Now, let $y\in G_{N}^{\varepsilon}$ and let $Q\in[\sqrt{\varepsilon}N,N]$. Then, for all $k=1,\dots,n$, by definition of $E_{\varepsilon,N}^{k}$ and our choice of $\beta$,

[TABLE]

Hence, in particular, for any $y\in\limsup_{N}G_{N}^{\varepsilon}$, there exists a sequence $N_{i}\rightarrow\infty$ for which (3.2) holds for all $Q\in[\sqrt{\varepsilon}N_{i},N_{i}]$ and for all $k=1,\dots,n$. Define the following sequence for $y\in\limsup_{N}G_{N}^{\varepsilon}$

[TABLE]

Now, a simple computation shows that for all $N$, and any function $\psi$,

[TABLE]

Combining 3.2 and 3.2 implies that for every $y\in\limsup_{N}G_{N}^{\varepsilon}$, all $k\leq n$ and for all $Q\in A(y,\varepsilon)$ such that $Q\gg M$,

[TABLE]

Note that the choice of $\varepsilon$ in the above considerations was arbitrary. Hence, we may consider sets $B^{\varepsilon_{i}}=X\setminus\limsup_{N}G_{N}^{\varepsilon_{i}}$, where $\varepsilon_{i}=1/i^{2}$. Then, by (3.4) and the Borel-Cantelli lemma applied to $B^{\varepsilon_{i}}$,

[TABLE]

Let $y\in X\setminus\limsup_{i}B^{\varepsilon_{i}}$. We claim that there exists a sequence $A(y)\subseteq\mathbb{N}$ of full upper density, such that for all $k\leq n$,

[TABLE]

By (3.9), this will conclude the proof.

Since $y\in\limsup_{N}G_{N}^{\varepsilon_{i}}$ for all $i$ sufficiently large, we have sequences $A(y,\varepsilon_{i})$ as before for all $\varepsilon_{i}$ sufficiently small. Without loss of generality, we may assume this holds for all $\varepsilon_{i}$. Note that by (3.6), the upper density of $A(y,\varepsilon_{i})$ is at least $\frac{1-\sqrt{\varepsilon_{i}}}{1+\varepsilon_{i}/C}$.

We build the sequence $A(y)$ from $A(y,\varepsilon_{i})$ by induction as follows. Let $N_{i}\in A(y,\varepsilon_{i})$ be such that (3.8) holds for all $k\leq n$ and all $Q\geq N_{i}$. Let $M_{1}=N_{1}$. If $M_{j}$ is defined, let $M_{j+1}\in\mathbb{N}$ be such that

[TABLE]

This in particular, implies that

[TABLE]

Now, define the sequence $A(y)$ as follows:

[TABLE]

Thus, since $\varepsilon_{j}\rightarrow 0$, the upper density of $A(y)$ is equal to $1$. Moreover, by (3.8) and by choice of $M_{j}$, we have that (3.10) holds as desired.

4. An Analogue of Birkhoff’s Theorem for Sequences of Transformations

In this section, we prove Theorem 1.2. We will use similar ideas to those used in the proof of Theorem 1.1 by applying the weak-type maximal inequality to a carefully chosen set of continuous functions capturing the structure of the ergodic invariant measures under the transformation $S$.

First, we make some standard reductions. Note that since $X$ is locally compact and second countable, by passing to the one point compactification and extending all the transformations on $X$ trivially to the point at infinity, we may assume that $X$ is in fact compact.

The set $Z$ in the assumption will then be enlarged to include the point at infinity since the dirac measure at that point will be an ergodic invariant measure for $S$. Also, since $X$ is now assumed compact, the space of probability measures on it is weak-$\ast$ compact and thus we can always find limit points of infinite sequences.

Proof of Theorem 1.2.

Let $\varepsilon\in(0,1)$ be fixed and let $\varepsilon_{n}=\varepsilon^{2}/4^{n}$ for $n\in\mathbb{N}$. Write $Z=\cup_{n}K_{n}$, where $K_{n}$ is compact and $K_{n}\subseteq K_{n+1}$ for each $n$. By regularity of the measure $\mu$, since $\mu(Z)=0$, there exists an open set $U_{n}$ containing $Z$, such that $\mu(U_{n}\setminus K_{n})<\varepsilon_{n}$, for each $n$.

Moreover, by Urysohn’s lemma, we can find a continuous function $0\leq f_{n}\leq 1$ such that $f_{n}|_{K_{n}}\equiv 1$ and $f_{n}\equiv 0$ on $X\setminus U_{n}$. Thus,

[TABLE]

Let $n$ be fixed. For each $j\in\mathbb{N}$, $k\leq n$ and $\alpha\in\mathbb{R}$, define the following set

[TABLE]

Applying the analogue of the maximal inequality, Proposition 2.1, with $f_{k}$, $\alpha_{k}=\beta_{k}=\varepsilon_{k}^{1/4}$, we get that for all $N$ sufficiently large, depending on $f_{k}$, there exists $j_{N,k}\in[0,\beta_{k}N]$ such that

[TABLE]

for each $k\leq n$.

Let $G_{N,k}=X\setminus E_{\alpha_{k},N,j_{N,k}}^{k}$. Let $y\in G_{N,k}$ and let $Q\in[\varepsilon_{k}^{1/8}N,N]$. Then, by definition of $E_{\alpha_{k},N,j_{N}}^{k}$,

[TABLE]

Now, for each $N\gg 1$, depending on $n$, define the following set

[TABLE]

and let $W_{n}=\limsup_{N}V_{N,n}$. The sets $W_{n}$ have the following properties:

• •

By (4.1), since $\nu(G_{N,k})\geq 1-\varepsilon/2^{k}$, $k=1,\dots,n$, we have $\nu(V_{N,n})\geq 1-\varepsilon$ for all $N\gg 1$. In particular,

[TABLE]

• •

For each $y\in W_{n}$, by (4) and (4.3), and noting that $\varepsilon>\varepsilon_{k}$, there exists a sequence $A(y,n)\subseteq\mathbb{N}$ defined by

[TABLE]

such that for all $k=1,\dots,n$ and for all $Q\in A(y,n)$,

[TABLE]

Let $W=\limsup_{n}W_{n}$. Then, by (4.4),

[TABLE]

Let $y\in W$. Then, there exists a sequence $n_{i}\rightarrow\infty$, such that $y\in W_{n_{i}}$ for all $i$. We will construct a sequence $A(y)$ from the sequences $A(y,n_{i})$ defined in (4.5) as follows. Let

[TABLE]

First, we define a sequence $N_{i}$ by induction. Let $N_{0}=1$. If $N_{j}$ has been defined, let $N_{j+1}$ be such that the following holds

[TABLE]

This is possible since the sequences $A(y,n)$ have upper density at least $1-\eta$. These conditions imply that

[TABLE]

Now, define the sequence $A(y)$ as follows:

[TABLE]

Thus, by (4.8), we get

[TABLE]

We claim that

[TABLE]

Let $\lambda_{\infty}^{y}$ be any weak-$\ast$ limit of the sequence $\frac{1}{N}\sum_{n=1}^{N}\delta_{T_{n}y}$, $N\in A(y)$. First, we claim that $\lambda_{\infty}^{y}(Z)=0$. Suppose otherwise. Then, since $Z=\bigcup_{i}K_{i}$ and $K_{i}\subseteq K_{i+1}$ for all $i$, there exists some $i_{0}$ such that for all $i>i_{0}$:

[TABLE]

Fix some $i>i_{0}$. By definition of the functions $f_{i}$, $\lambda_{\infty}^{y}(f_{i})\geq\lambda_{\infty}^{y}(K_{i})$. Then, for all $n_{j}\geq i$ and for all $N\in A(y,n_{j+1})\cap[N_{j},N_{j+1}]\subset A(y)$, by (4.6), we get

[TABLE]

In particular, we get that $\lambda_{\infty}^{y}(K_{i_{0}})\leq 2\varepsilon_{i}^{1/8}$. But, this applies to $i>i_{0}$. Thus, since $\varepsilon_{i}\rightarrow 0$, we get that $\lambda_{\infty}^{y}(K_{i_{0}})=0$, a contradiction.

Next, by our hypothesis, (after possibly intersecting $W$ with a set of full measure), $\lambda_{\infty}^{y}$ is $S$-invariant. However, all the ergodic $S$-invariant measures different from $\mu$ live on $Z$ to which $\lambda_{\infty}^{y}$ assigns [math] mass. Thus, by the ergodic decomposition, we get that $\lambda_{\infty}^{y}=\mu$. Hence, the sequence $\frac{1}{N}\sum_{n=1}^{N}\delta_{T_{n}y}$, $N\in A(y)$ has $\mu$ as its only weak-$\ast$ limit point as desired.

Thus far, we proved that for all $y\in W$, a set of $\nu$ measure at least $1-\varepsilon$, there exists a sequence $A(y)$ of upper density at least $1-3\eta$ such that (4.11) holds. Since $\varepsilon$ was arbitrary, the conclusion of the theorem holds $\nu$ almost everywhere as desired.

∎

5. Existence of Ratner Sequences

In this section, we prove a general criterion for the existence of Ratner sequences, Theorem 1.4. Let us fix some notation. Let $G$ be a semisimple Lie group and let $H$ be a closed connected Lie subgroup of $G$. Let $\Gamma$ be a lattice in $G$ and assume that the orbit $H\Gamma$ is closed in $G/\Gamma$ and supports an $H$-invariant probability measure $\mu_{H}$. Let $Z_{G}(H)$ denote the centralizer of $H$ in $G$. Let $\mathfrak{g}$ and $\mathcal{h}$ denote the Lie algebras of $G$ and $H$ respectively. For $g\in G$, let $Ad(g)$ denote the linear transformation on the Lie algebra of $G$ induced by the adjoint action of $g$. Let $\left\lVert\cdot\right\rVert$ denote some

Recall that a sequence of elements $g_{n}\in G$ is said to be a Ratner sequence if there exists a one-parameter unipotent subgroup $W<G$ such that for $\mu_{H}$-almost every $x\in G/\Gamma$, any limit point of the empirical measures $\frac{1}{N}\sum_{n=1}^{N}\delta_{g_{n}x}$ is invariant by $W$.

5.1. $SL_{2}(\mathbb{R})$ Representations and Unipotent Invariance

The first key step in the proof of Theorem 1.4 is Proposition 5.1 below. It relies on the representation theory of embedded copies of $SL_{2}(\mathbb{R})$ inside semisimple Lie groups and its proof is inspired by Ratner’s H-principle appearing in the proof of her measure classification theorem.

The statement roughly says that it is possible to change the starting point of a unipotent orbit of a group $U$ in a direction parallel to $H$ so that the $2$ unipotent orbits differ roughly by a unipotent element in the centralizer of $U$. The following is the precise statement.

Proposition 5.1**.**

Let $U=\left\{u_{t}:t\in\mathbb{R}\right\}$ be an $Ad$-unipotent $1$-parameter subgroup of $G$ such that $U\not\subset Z_{G}(H)$. Then, for every sequence $t_{n}\rightarrow\infty$, there exists a sequence $v_{n}\in\mathfrak{h}=Lie(H)$ satisfying the following: for all $n$ sufficiently large

[TABLE]

where $u\in Z_{G}(U)$ is a non-trivial $Ad$-unipotent element of $G$.

Proof.

Let $X\in\mathfrak{g}$ be such that $u_{t}=\exp(tX)$ for all $t\in\mathbb{R}$. Since $u_{t}$ is $Ad$-unipotent, $X$ is an $ad$-nilpotent element of $\mathfrak{g}$. That, $ad(X)$ is a nilpotent linear transformation of $\mathfrak{g}$.

Since $\mathfrak{g}$ is a semisimple Lie algebra, by the Jacobson-Morozov theorem, $X$ may be extended to an $\mathfrak{sl}_{2}$-triple. That is there exists $Y,H\in\mathfrak{g}$ such that the following relations hold:

[TABLE]

Hence, the Lie subalgebra $\mathfrak{f}$ generated by $X,Y$ and $H$ is isomorphic to the Lie algebra $\mathfrak{sl}_{2}(\mathbb{R})$. Since $\mathfrak{f}$ is semisimple, $\mathfrak{g}$ decomposes into irreducible representations under the adjoint action of $\mathfrak{f}$ as follows:

[TABLE]

For $1\leq i\leq s$, let $\pi_{i}:\mathfrak{g}\rightarrow V_{i}$ denote the associated projections and note that $Ad(u_{t})$ commutes with $\pi_{i}$ for all $i$. Let $v\in\mathfrak{h}$.

Let $1\leq i\leq s$ be such that $\pi_{i}(v)$ is not fixed by $Ad(u_{t})$. Let $n_{i}\in\mathbb{N}$ be such that

[TABLE]

By the standard description of irreducible $\mathfrak{sl}_{2}(\mathbb{R})$ representations, $V_{i}$ decomposes into $1$ dimensional eigenspaces for the action $H$ as follows:

[TABLE]

so that for each $0\leq l\leq n_{i}$ and every $w\in W_{l}^{(i)}$, we have

[TABLE]

Let $q_{l}:V_{i}\rightarrow W_{l}^{(i)}$ denote the associated projections. Let $\left\{w_{l}^{(i)}:0\leq l\leq n_{i}\right\}$ denote a basis of $V_{i}$ consisting of eigenvectors of $H$ and write

[TABLE]

Note that for each $l$, we have that

[TABLE]

In particular, we get the following

[TABLE]

Note also that the degree of the polynomial appearing in the coefficient of $q_{l}(\pi_{i}(Ad(u_{t})(v)))$ for any $l>0$ is strictly less than the degree of the polynomial in (5.1). Let

[TABLE]

And, define the following natural number

[TABLE]

By assumption, we have that $U\not\subset Z_{G}(H)$. Thus, $d_{\mathfrak{h}}\neq 0$. In particular, we can find $v\in\mathfrak{h}$ and $1\leq i\leq s$ such that

[TABLE]

Now, let $t_{n}\in\mathbb{R}\setminus\left\{0\right\}$ be a sequence tending to infinity and define $v_{n}$ to be

[TABLE]

Thus, for each $i$ with $d_{i}(v)=d_{\mathfrak{h}}$, by (5.1), we get that

[TABLE]

And, for each $l>0$, we have that

[TABLE]

In particular, we get that

[TABLE]

Next, since $v_{n}\rightarrow 0$, $\exp(v_{n})$ converges to the identity element in $G$. Hence, all the eigenvalues of the linear transformation $Ad(\exp(v_{n}))$ tend to $1$. Thus, since conjugation doesn’t change eigenvalues, we get that $Ad(u_{t_{n}})(\exp(v_{n}))$ converges to an $Ad$-unipotent element of $G$ which is non-trivial by (5.2). Since $Ad(u_{t})$ fixes $w_{0}^{(i)}$ for all $i$, by (5.2), the limiting element belongs to the centralizer of $U$.

Finally, note that equation (5.1) and the definition of $d_{\mathfrak{h}}$ imply that for all $t>0$ sufficiently large

[TABLE]

This completes the proof. ∎

The following example demonstrates the above Proposition in the concrete set up of $G=SL_{2}(\mathbb{R})$.

Example 5.2**.**

Let $G=SL(2,\mathbb{R})$ and $H=K=SO(2)$. Let $t_{n}\rightarrow+\infty$ be a sequence. Let $g_{n}$ be the following sequence:

[TABLE]

Let $k_{\theta}\in K$. Then,

[TABLE]

Let $\alpha\in(0,1)$ be a fixed real number. For all large $n$, let $\theta_{n}$ be such that $t_{n}^{2}\sin(\theta_{n})=\alpha$. Then, as $n\rightarrow\infty$, $\theta_{n}\rightarrow 0$ and $t_{n}\sin(\theta_{n})\rightarrow 0$. Hence, we get

[TABLE]

5.2. Decay of Correlations

Let $\Gamma$, $H$ and $G$ be as above and define

[TABLE]

Let $\varphi\in C_{c}^{\infty}(G/\Gamma)$. For each $n$, define the following function on $H\Gamma/\Gamma$:

[TABLE]

where $v_{n}\in Lie(H)$ is as in the conclusion of Proposition 5.1 applied to the sequence $t_{n}$. The reason for defining such functions is the following

Proposition 5.3**.**

To prove Theorem 1.4, it suffices to show that for $\mu_{H}$ almost every $x\in H\Gamma/\Gamma$, the following holds:

[TABLE]

Proof.

Let $h_{n}=\exp(v_{n})$. By Proposition 5.1, we have $g_{n}h_{n}g_{n}^{-1}\rightarrow u$, where $u$ is a non-trivial Ad-unipotent element. Since, $\varphi$ is uniformly Lipschitz, we have

[TABLE]

where $d(.,.)$ is the right invariant metric on $G$. Hence, $|\varphi(ug_{n}x)-\varphi(g_{n}h_{n}x)|\rightarrow 0$ as $n\rightarrow\infty$.

Hence, by assumption,

[TABLE]

But, $\varphi$ was an arbitrary function. Thus, any limit point must be invariant by the group generated by $u$.

∎

In order to use Proposition 5.3, we shall need the following estimate on the correlation between the functions $f_{n}$. This lemma is an adaptation of the key technical lemma in [CE] (Lemma 3.3) to our setting. we shall need the following definition.

Definition 5.4**.**

For any $x\in H\Gamma/\Gamma\cong H/(H\cap\Gamma)$, the injectivity radius at $x$, denoted by $inj_{x}$ is defined to be the infimum over all $r>0$ such that the map $h\mapsto hx$ is injective on the ball of radius $r$ around identity in $H$.

Lemma 5.5**.**

For all $n\geq m\geq 1$ such that $\left\lVert Ad(g_{m})|_{\mathfrak{h}}\right\rVert/\left\lVert Ad(g_{n})|_{\mathfrak{h}}\right\rVert$ is sufficiently small, the following holds

[TABLE]

Proof.

Let $v_{n}$ be as in the definition of the functions $f_{n}$ and let $h_{n}=\exp(v_{n})$. Let $d_{n}=\left\lVert Ad(g_{n})|_{\mathfrak{h}}\right\rVert$ and $d_{m}=\left\lVert Ad(g_{n})|_{\mathfrak{h}}\right\rVert$. Define

[TABLE]

Let $B_{H}(e,r)$ denote the ball of radius $r$ around the identity in $H$. By abuse of notation, we’ll use $\mu_{H}$ to denote the Haar measure on $H$ and on $H\Gamma/\Gamma$.

Let $\psi:H\Gamma/\Gamma\rightarrow\mathbb{R}$ be any integrable function. Then, by Fubini’s theorem and left $H$-invariance of $\mu_{H}$,

[TABLE]

Define the following set

[TABLE]

where $inj_{x}$ denotes the injectivity radius at $x$ in $H/(H\cap\Gamma)$. Let $Thin_{r}=H\Gamma/\Gamma-Thick_{r}$. Using the structure of Siegel sets, one can show (Lemma 11.2, [BO]) that $\mu_{H}(Thin_{r})\ll r^{p}$, for some $p>0$ as $r\rightarrow 0$. Hence, it suffices to prove for all $x\in Thick_{r}$,

[TABLE]

Let $w\in Thick_{r}$ be fixed. Let $B_{r}$ denote the ball of radius $r$ around the $w$ in $H\Gamma/\Gamma$ in the metric induced by the metric on $G$. Then, for every $x\in B_{r}$, there exists some $l\in B_{H}(e,r)$ such that $x=lw$. Since $\varphi\in C_{c}^{\infty}(G/\Gamma)$, $\varphi$ is uniformly Lipschitz. Thus, we get

[TABLE]

Since the sequences $d_{n},d_{m}$ are tending to infinity, for all $n,m$ sufficiently large, $r$ will be small enough so that the exponential map is a diffeomorphism from a neighborhood of [math] in $\mathfrak{h}=Lie(H)$ onto $B_{H}(e,r)$.

Thus, we can write $l=\exp(v)$ for some $v\in\mathfrak{h}$. So, we have

[TABLE]

where for any $g\in G$.

But, since $h_{m}\rightarrow id$ as $m\rightarrow\infty$ and since the norm is continuous, for all $m$ sufficiently large, we have $\left\lVert Ad(h_{m})\right\rVert\ll 1$.

Moreover, since the differential of the exponential map at [math] is the identity, its Jacobian is $1$ at [math] and hence, when $r$ is sufficiently small, we have $\left\lVert v\right\rVert\ll d(l,e)\leq r$. Combining these estimates, we get for all $x\in B_{r}$,

[TABLE]

But, as before, the exponential map is nearly an isometry near identity. Hence, when $\left\lVert Ad(g_{n})|_{\mathfrak{h}}\right\rVert$ is sufficiently larger than $\left\lVert Ad(g_{m})|_{\mathfrak{h}}\right\rVert$, $Ad(g_{m}h_{m})(v)$ will be sufficiently close to [math] so that $d(\exp(Ad(g_{m}h_{m})(v)),e)\ll\left\lVert Ad(g_{m}h_{m})(v)\right\rVert$ up to absolute constants. Thus, we get for all $x\in B_{r}$,

[TABLE]

Similarly, we get the same estimate for $\varphi(g_{m}x)-\varphi(g_{m}w)$ for all $x\in B_{r}$. Thus, by definition of $f_{m}$, we get

[TABLE]

Thus, we get that

[TABLE]

Next, note that by definition of $f_{n}$ and left-invariance of $\mu_{H}$,

[TABLE]

Since $w\in Thick_{r}$, $B_{r}$ isometric to $B_{H}(e,r)$. Hence, for all $r$ sufficiently small, we may apply Proposition 5.6 below to get

[TABLE]

Note that Proposition 5.6 requires that $h_{n}\in B_{H}(e,r)$. To get around this assumption, observe that for $n$ sufficiently large, $h_{n}=\exp(v_{n})$ will be sufficiently close to identity and hence, we have

[TABLE]

But, by Proposition 5.1, we have $\left\lVert v_{n}\right\rVert\ll 1/\left\lVert Ad(g_{n})|_{\mathfrak{h}}\right\rVert$ for all $n$ sufficiently large. But, since $\left\lVert Ad(g_{n})|_{\mathfrak{h}}\right\rVert\geq\left\lVert Ad(g_{m})|_{\mathfrak{h}}\right\rVert$ by assumption, we have $\left\lVert Ad(g_{n})|_{\mathfrak{h}}\right\rVert\geq 1/r$. Thus, in particular, $h_{n}$ will be contained in a ball of radius comparable to $r$ for all large $n$, which doesn’t affect our estimate.

Moreover, this observation, along with (5.9), imply that

[TABLE]

Combining this estimate with (5.7) and (5.8) gives

[TABLE]

and the conclusion of the lemma follows. ∎

5.2.1. A measure estimate

The following estimate was used in the proof of Lemma 5.5.

Proposition 5.6**.**

Let $H$ be a Lie group and let $B_{r}$ denote a ball of radius $r>0$ around the identity in $H$. Then, for all $r>0$ sufficiently small and all $h\in B_{r}$,

[TABLE]

where $\mu_{H}$ denotes a left-invariant Haar measure on $H$ and $d(.,.)$ denotes a right invariant metric.

Proof.

Let $\mathfrak{h}=Lie(H)$. Fix a norm on $\mathfrak{h}$ inducing the metric $d$. Let $r>0$ be small enough such that the exponential map is a diffeomorphism from a ball around [math] in $\mathfrak{h}$ onto $B_{r}$. Since the differential of the exponential is the identity at [math], such ball will have a radius comparable to $r$, denote it by $B_{r^{\prime}}^{\mathfrak{h}}$

Let $g\in B_{r}$. Let $X,Y\in\mathfrak{h}$ be such that $h=\exp(X)$ and $g=\exp(Y)$. Then, if $r$ is sufficiently small, by the Campell-Baker-Hausdorff formula, there exists some $Z\in\mathfrak{h}$ so that $hg=\exp(Z)$ and

[TABLE]

In particular, there is some $C\geq 1$ such that $hB_{r}\subseteq\exp(B_{r^{\prime}}^{\mathfrak{h}}+CX)$. And, hence, we get that

[TABLE]

Let $Leb$ denote the Lebesgue measure on $\mathfrak{h}$. It is then a standard fact from convex euclidean geometry that

[TABLE]

where the implicit constants are absolute and depend only on the dimension (see for example [Gro]). Here we are using that a ball in the norm on $\mathfrak{h}$ is equivalent to a standard euclidean ball of comparable radius.

Again, since the differential of the exponential is the identity at [math], the Haar measure on on $H$ near identity is comparable up to absolute constants with the pushforward of the Lebesgue measure under the exponential map.

In particular, one has $\mu_{H}(B_{r})\ll r^{dimH}$. Combining this with (5.12) gives the desired conclusion. ∎

5.3. Law of Large Numbers

This section is dedicated to the proof of Proposition 5.7 below. The proof follows the same ideas as in the proof of the strong law of large numbers and is very similar to the proof in [CE]. The main difference is the need to handle the general growth condition in Theorem 1.4.

Recall the definition of the functions $f_{n}$ from the previous section. By Proposition 5.3, the following proposition concludes the proof of Theorem 1.4.

Proposition 5.7**.**

Under the same hypotheses of Theorem 1.4, for $\mu_{H}$ almost every $x\in H\Gamma/\Gamma$,

[TABLE]

Proof.

For $x\in H\Gamma/\Gamma$ and $N\in\mathbb{N}$, let $S_{N}(f)(x)=\sum_{n=1}^{N}f_{n}(x)$. As noted in equation (5.3), there exists a natural number $d\geq 1$, depending only on $H$ and $U$, such that

[TABLE]

as $n$ tends to infinity. Hence, by assumption, there exists $\lambda>0$ such that for all $n\geq 1$, we have

[TABLE]

Then, we have

[TABLE]

Here we estimated the number of pairs $(m,n)$ with $|m-n|<N^{1/2}$ using the area between the $2$ lines $m\pm n=N^{1/2}$ in the square $[0,N]^{2}$.

But, by (5.13), when $N\gg 1$, for $n\geq m$ such that $|n-m|\geq N^{1/2}$, we have that $\left\lVert Ad(g_{m})|_{\mathfrak{h}}\right\rVert/\left\lVert Ad(g_{n})|_{\mathfrak{h}}\right\rVert$ will be sufficiently small so that Lemma 5.5 applies. This implies that for all $N\gg 1$:

[TABLE]

Let $\varepsilon>0$. Then, by the Chebyshev-Markov inequality,

[TABLE]

For all $k\in\mathbb{N}$, let $N_{k}=k^{4}$. Thus, the above observation shows that the sequence $N_{k}^{-1/2}$ is summable. Hence, by the Borel-Cantelli lemma, we have

[TABLE]

Since $\varepsilon$ was arbitrary, by taking a countable sequence $\varepsilon_{i}$ decreasing to [math], we conclude that for $\mu_{H}$ almost every $x$,

[TABLE]

We are left with bootstrapping this conclusion to all sequences, for which we use a standard interpolation argument. Let $M_{i}\rightarrow\infty$ be a sequence. Observe that for each $M_{i}\in\mathbb{N}$, there exists some $k_{i}\in\mathbb{N}$ such that $N_{k_{i}}\leq M_{i}\leq N_{k_{i}+1}$.

Moreover, we have $N_{k_{i}+1}-N_{k_{i}}=O(k_{i}^{3})$. Thus, we get that

[TABLE]

as desired. ∎

6. Proof of Theorem 1.3

This section is dedicated to the proof of Theorem 1.3. Let the notation be the same as in § 5. Recall that a sequence $g_{n}$ of elements of $G$ is said to be a Ratner Sequence if there exists a non-trivial $Ad$-unipotent element $u\in G$ such that for $\mu_{H}$ almost every $x\in G/\Gamma$, any limit point of the sequence of empirical measures $\frac{1}{N}\sum_{n=1}^{N}\delta_{g_{n}x}$ is invariant by $u$.

The existence of such sequences is the content of Theorem 1.4.

Proof of Theorem 1.3.

Let $g_{n}$ be a Ratner sequence for $H$. Let $W$ denote the one-parameter unipotent subgroup generated by an $Ad$-unipotent element $u$ as in the definition of Ratner sequences above.

We will apply Theorem 1.2 with $X=G/\Gamma$, $\mu=\mu_{G/\Gamma}$, $\nu=\mu_{H}$ and $T_{n}=g_{n}$. Let us verify the hypotheses. By assumption, we have that

[TABLE]

In particular, this implies condition $(1)$ of Theorem 1.2. Moreover, by definition of Ratner sequences, condition $(2)$ is satisfied, with the transformation $S$ being multiplication by the unipotent element $u$.

Let $\mathcal{L}$ denote the collection of proper analytic subgroups $L$ of $G$ such that $L\cap\Gamma$ is a lattice. Then, $\mathcal{L}$ is a countable set [Rat].

For $L\in\mathcal{L}$, define the following set

[TABLE]

Let $\pi:G\rightarrow G/\Gamma$ denote the natural projection. The set $Z$ appearing in the hypotheses of Theorem 1.2 will be defined to be

[TABLE]

Then, since $Z$ is a countable union of analytic subvarieties of $G/\Gamma$ [Rat], $Z$ admits a filtration by compact sets. Moreover, since $\mathcal{L}$ is countable, and $\mu_{G/\Gamma}(\pi(N(L,W)))=0$ for all $L\in\mathcal{L}$, we have

[TABLE]

Finally, by Ratner’s measure rigidity theorem [Rat], any ergodic $W$ invariant probability measure $\lambda\neq\mu_{G/\Gamma}$ is supported on $N(L,W)$ (in fact supported on a single closed orbit of a conjugate of $L$) for some $L\in\mathcal{L}$. Thus, all the hypotheses of Theorem 1.2 are verified and hence the conclusion of Theorem 1.3 follows.

∎

Appendix A Symmetric Groups and Ratner Sequences

Throughout this section, $G$ is a connected semisimple Lie group with finite center and $H$ is symmetric subgroup of $G$. We let $\Gamma$ be an irreducible lattice in $G$ and assume the orbit $H\Gamma$ is closed in $G/\Gamma$ and supports an $H$-invariant probability measure.

We prove a general criterion for a sequence of elements of a Lie group $G$ to contain a Ratner sequence as a subsequence with respect to $H$. The precise statement is Theorem A.1 below. The proof of this theorem follows the same lines as the proof of Theorem 1.4. The only difference being Proposition A.3 below which acts as a replacement for Proposition 5.1 in the proof of Theorem 1.4. The arguments in sections 5.2 and 5.3 carry over verbatim to this setting.

A.1. Structure of Affine Symmetric Spaces

Our main tool will be the structure of affine symmetric spaces which we recall here. We follow the exposition in [EM] closely for the material in this section. Let $\sigma:G\rightarrow G$ be an involution such that $H$ is the fixed point set of $\sigma$. Then, $G/H$ is an affine symmetric space. By abuse of notation, let $\sigma$ also denote the differential of $\sigma$ at identity. Let $\mathfrak{g}$ denote the Lie algebra of $G$. Then, we have

[TABLE]

where $\mathfrak{h}$ is the eigenspace corresponding to the eigenvalue $1$ of $\sigma$ and $\mathfrak{p}$ corresponds to the $-1$ eigenspace, and $\mathfrak{h}$ is the Lie algebra of $H$.

It is well known (Proposition 7.1.1, [Sch]) that one can find a Cartan involution $\theta$ of $G$ commuting with $\sigma$. Let $\theta$ also denote its differential at identity. Then, similarly $\mathfrak{g}$ splits as

[TABLE]

where $\mathfrak{k}$ (resp. $\mathfrak{q}$) is the $+1$ (resp. $-1$) eigenspace of $\theta$. Since $\theta$ is a Cartan involution, $\mathfrak{k}$ is the Lie algebra of a maximal compact subgroup, denote it by $K$.

Now, let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}\cap\mathfrak{q}$. Then, $\mathfrak{a}$ is the Lie algebra of a maximal abelian subgroup $A$ and the exponential map $\mathfrak{a}\rightarrow A$ is a diffeomorphism.

Recall that $G$ admits a decomposition of the form $G=KAH$ (See [Sch], Proposition 7.1.3 or [EM], Proposition 4.2). Elements of the fiber of the map $(k,a,h)\mapsto kah$ have the form $(kl,a,l^{-1}h)$ for some element $l\in K\cap H$. In particular, the fiber lies in a compact group.

Consider the adjoint action of $\mathfrak{a}$ on $\mathfrak{g}$. There exists a finite subset $\Sigma\subset{\mathfrak{a}^{\ast}}$ of non-zero elements of the dual of $\mathfrak{a}$ such that $\mathfrak{g}$ splits as

[TABLE]

such that for all $X\in\mathfrak{a}$ and all $Z\in\mathfrak{g}_{\alpha}$,

[TABLE]

And, for $Z\in\mathfrak{g}_{0}$, $ad_{X}(Z)=0$. Recall that the subspaces

[TABLE]

for $\alpha\in\Sigma$ divide $\mathfrak{a}$ into a finite collection of cones, called Weyl Chambers.

Let $\mathfrak{C}$ be one such Weyl chamber. Let $\Sigma^{+}$ denote the set of $\alpha\in\Sigma$ such that $\alpha(X)>0$ for all $X\in\mathfrak{C}$. We call $\Sigma^{+}$ the set of positive roots relative to $\mathfrak{C}$. Then, $\mathfrak{g}$ splits as follows:

[TABLE]

where $\mathfrak{n}^{-}=\bigoplus_{\alpha\in\Sigma-\Sigma^{+}}\mathfrak{g}_{\alpha}$ and $\mathfrak{n}^{+}=\bigoplus_{\alpha\in\Sigma^{+}}\mathfrak{g}_{\alpha}$.

A.2. Unipotent Invariance

We need to fix some notation before stating the main theorem. Fix some norm on $Lie(G)$ inducing the metric on $G$ and denote it by $\left\lVert\cdot\right\rVert$. This norm induces a matrix norm for the Adjoint maps. Let $\left\lVert Ad(g)\right\rVert$ denote such matrix norm.

The following is the main theorem of this section.

Theorem A.1**.**

In the notation above, if $g_{n}\in G$ is a sequence tending to infinity in $G/H$ and satisfying the following growth condition:

1. (1)

There exists a constant $\lambda>0$ such that for all $n\geq 1$,

[TABLE] 2. (2)

Writing $g_{n}=k_{n}a_{n}h_{n}$, we have that $\left\lVert Ad(h_{n}^{-1})\right\rVert$ is uniformly bounded for all $n$.

Then, $g_{n}$ contains a Ratner sequence for $H$ as a subsequence.

Remark A.2**.**

1. (1)

Passage to a subsequence in the conclusion of Theorem A.1 is needed to insure invariance in the limit by a single one-parameter unipotent subgroup. This is very important for applying Theorem 1.2 to prove Theorem 1.3. 2. (2)

The second growth condition in Theorem A.1 makes sense, since the element $h_{n}$ in the decomposition of $g_{n}$ is unique up to left multiplication by elements inside the compact group $H\cap K$. 3. (3)

The growth rate of $\left\lVert Ad(g_{n})\right\rVert$ required by this theorem is not the most general one which works with our techniques. It is possible to obtain the same conclusions assuming there exist constants $\lambda,c>0$ such that $\left\lVert Ad(g_{n})\right\rVert=O\left(e^{\lambda n^{c}}\right)$.

A.3. Expansion Properties of the Adjoint Action

We shall need the following lemma regarding the Adjoint action of $G$. This lemma exploits the relationship between diagonalizable elements and their associated horospherical subgroups. We also make use of the structure of affine symmetric spaces.

Proposition A.3**.**

Let $g_{n}$ be as in Theorem A.1. Then, there exists a sequence $v_{n}\rightarrow 0\in Lie(H)$ satisfying the following for all $n$,

[TABLE]

and such that after passing to a subsequence of the $g_{n}$’s, we have

[TABLE]

where $u$ is an Ad-unipotent element in $G$.

We will need the following fact for the proof of Proposition A.3.

Lemma A.4**.**

(Lemma 3, [Moz]) If $G$ is semisimple over $\mathbb{R}$ with finite center, then the Adjoint representation $Ad:G\rightarrow GL(\mathfrak{g})$ is a proper map.

We are now ready for the proof.

A.3.1. Proof of Proposition A.3

Write $g_{n}=k_{n}a_{n}h_{n}$. Then, by passing to a subsequence of $g_{n}$, we may assume that there exists a single Weyl chamber $\mathfrak{C}$ such that $a_{n}=\exp(X_{n})$ and $X_{n}\in\mathfrak{C}$ for all $n$. Let $\Sigma^{+}$ be a set of positive roots associated with $\mathfrak{C}$.

First, we’ll assume that $g_{n}\in KA$ and write $g_{n}=k_{n}a_{n}$. Note that the map $Ad:G\rightarrow GL(\mathfrak{g})$ is proper by Lemma A.4. In particular, by assumption, since $g_{n}\rightarrow\infty$, we have

[TABLE]

We claim that the image of $\mathfrak{h}$ under the projection onto $\mathfrak{n}^{+}$ is non-zero. To see this, note that given $X,Y\in\mathfrak{p}$, we have that

[TABLE]

Hence, $ad_{X}(Y)\in\mathfrak{h}$. On the other hand, if $X\in\mathfrak{a}\subseteq\mathfrak{p}$ and $Y\in\mathfrak{g}_{\alpha}$ for some $\alpha\neq 0\in\Sigma$, then $ad_{X}(Y)\in\mathfrak{g}_{\alpha}$. In particular, this implies

[TABLE]

Thus, given any $X\neq 0\in\mathfrak{n}^{+}$, the element $X+\sigma(X)\neq 0$ and is $\sigma$ invariant and hence belongs to $\mathfrak{h}$.

Next, note that since $\sigma(X)=-X$ for all $X\in\mathfrak{a}$, we have $\sigma(\mathfrak{n}^{+})=\mathfrak{n}^{-}$. Thus, in particular, for any $v\in\mathfrak{n}^{+}$,

[TABLE]

Let $V=\left\{v_{\alpha}\in\mathfrak{g}_{\alpha}:\left\lVert v_{\alpha}\right\rVert=1,\alpha\in\Sigma^{+}\right\}$ be fixed. For each $n$, let $v_{\alpha_{n}}\in V$ be such that

[TABLE]

where $X_{n}\in\mathfrak{C}$ was such that $a_{n}=\exp(X_{n})$. Now, for each $n$, let

[TABLE]

Then, for all $n$, $v_{n}\neq 0$ in $\mathfrak{h}$ and satisfies

[TABLE]

Moreover, by the standard identity $Ad(\exp)=\exp(ad)$ and by A.1,

[TABLE]

By compactness of $K$, we have $\left\lVert Ad(g_{n})\right\rVert\ll\left\lVert Ad(a_{n})\right\rVert$. But, by our choice of $\alpha_{n}$, $e^{\alpha_{n}(X_{n})}$ is the largest eigenvalue of $Ad(a_{n})$ and $Ad(a_{n})$ is diagonalizable. Thus, $e^{\alpha_{n}(X_{n})}/\left\lVert Ad(a_{n})\right\rVert=O(1)$ and so we get

[TABLE]

Hence, by passing to a subsequence, we get that

[TABLE]

Since $v_{n}\rightarrow 0$, we have that $\exp(v_{n})\rightarrow id$. Hence, all the eigenvalues of $Ad(\exp(v_{n}))$ converge to $1$. Since conjugation doesn’t change eigenvalues, we get that $u$ must be an $Ad$-unipotent element, which finishes the proof in the case $g_{n}\in KA$.

For the general case, by $KAH$ decomposition, we write $g_{n}=k_{n}a_{n}h_{n}$. Then, we can find $v_{n}\in\mathfrak{h}$ as above such that $Ad(k_{n}a_{n})(\exp(v_{n}))\rightarrow u\neq id$. Thus, the elements $w_{n}=h_{n}^{-1}v_{n}h_{n}$ will satisfy $Ad(g_{n})(\exp(w_{n}))=Ad(k_{n}a_{n})(\exp(v_{n}))\rightarrow u$. By our assumption on the boundedness of $\left\lVert Ad(h_{n}^{-1})\right\rVert$, we get

[TABLE]

Hence, the sequence $w_{n}$ satisfies the conclusion of the Proposition.