2.1 Definitions

Let

be the set of $g\in SL_{n+1}(\mathbb {R})$ satisfying

$$ \begin{align*} \left[ \begin{array}{ccc} I_{n} & \\ & -1 \\ \end{array} \right]g^{T}\left[ \begin{array}{ccc} I_{n} & \\ & -1 \\ \end{array} \right]=g^{-1}, \end{align*} $$

where $I_{n}$ is the $n\times n$ identity matrix. The corresponding Lie algebra $\mathfrak {g}$ then consists of $v\in \mathfrak {sl}_{n+1}(\mathbb {R})$ satisfying

$$ \begin{align*}\left[ \begin{array}{ccc} I_{n} & \\ & -1 \\ \end{array} \right]v^{T}\left[ \begin{array}{ccc} I_{n} & \\ & -1 \\ \end{array} \right]=-v.\end{align*} $$

Then the Cartan decomposition can be given by

$$ \begin{align*}\mathfrak{g}=\mathfrak{l}\oplus\mathfrak{p}=\left\{\left[ \begin{array}{ccc} \mathbf{l} & \\ & 0 \\ \end{array} \right] :\mathbf{l}\in \mathfrak{so}(n) \right\}\oplus\left\{ \left[ \begin{array}{ccc} 0 & \mathbf{p} \\ \mathbf{p}^{T} & 0 \\ \end{array} \right]: \mathbf{p}\in\mathbb{R}^{n}\right\}\!.\end{align*} $$

Let $E_{ij}$ be the $(n\times n)$ -matrix with $1$ in the $(i,j)$ -entry and $0$ otherwise. Let $e_{k}\in \mathbb {R}^{n}$ be the kth standard basis (vertical) vector. Set

Then $Y_{i},\Theta _{ij}$ form a basis of $\mathfrak {g}=\mathfrak {so}(n,1)$ .

Let $\mathfrak {a}=\mathbb {R}Y_{n}\subset \mathfrak {p}$ be a maximal abelian subspace of $\mathfrak {p}$ . Then the root space decomposition of $\mathfrak {g}$ is given by

(2.1) $$ \begin{align} \mathfrak{g}=\mathfrak{g}_{-1}\oplus\mathfrak{m}\oplus\mathfrak{a}\oplus\mathfrak{g}_{1}. \end{align} $$

Denote by the sum of the positive root spaces. Let $\rho $ be the half sum of positive roots. We also adopt the convention by identifying $\mathfrak {a}^{\ast }$ with $\mathbb {C}$ via $\unicode{x3bb} \mapsto \unicode{x3bb} (Y_{n})$ . Thus, $\rho =\rho (Y_{n})=(n-1)/2$ .

Let $\Gamma \subset G$ be a lattice,

, $\mu $ be the Haar probability measure on X. Fix a nilpotent $U\in \mathfrak {g}_{-1}$ . On $G/\Gamma $ , denote by:

• • a geodesic flow;

• • a unipotent flow.

It is worth noting that

$$ \begin{align*}[Y_{n},U]=-U.\end{align*} $$

Then there exists $\overline {U}\in \mathfrak {g}$ such that $\{U,Y_{n},\overline {U}\}$ is an $\mathfrak {sl}_{2}$ -triple. Denote

For convenience, we choose

(2.2)

Then $\langle u^{t},a^{t},\overline {u}^{t}\rangle $ generates $\mathrm {SO}(2,1)\subset \mathrm {SO}(n,1)$ .

2.2 $\mathfrak {sl}_{2}$ -weight decomposition

First, consider an arbitrary Lie algebra $\mathfrak {g}$ as a $\mathfrak {sl}_{2}$ -representation via the adjoint map (after identifying an image of $\mathfrak {sl}_{2}$ by Jacobson– Morozov theorem), then by the complete reducibility of $\mathfrak {sl}_{2}$ , there is a decomposition of $\mathfrak {sl}_{2}$ -representations

(2.3) $$ \begin{align} \mathfrak{g}=\mathfrak{sl}_{2}\oplus V^{\perp}, \end{align} $$

where $V^{\perp }\subset \mathfrak {g}$ is the sum of $\mathfrak {sl}_{2}$ -irreducible representations other than $\mathfrak {sl}_{2}$ . In particular, for $\mathfrak {g}=\mathfrak {so}(n,1)$ , we have

(2.4) $$ \begin{align} V^{\perp}=\sum_{i} V_{i}^{0}\oplus\sum_{j} V_{j}^{2}, \end{align} $$

where $V_{i}^{0}$ and $V_{j}^{2}$ are $\mathfrak {sl}_{2}$ -irreducible representations with highest weights $0$ and $2$ . More precisely, we have the following lemma.

Lemma 2.1. By the weight decomposition, an irreducible $\mathfrak {sl}_{2}$ -representation $V^{\varsigma }$ is the direct sum of weight spaces, each of which is $1$ -dimensional. More precisely, there exists a basis $v_{0},\ldots ,v_{\varsigma }\in V^{\varsigma }$ such that

$$ \begin{align*}U.v_{i}=(i+1)v_{i+1},\quad Y_{n}.v_{i}= \frac{\varsigma-2i}{2}v_{i}.\end{align*} $$

Thus, if $V^{\varsigma }$ is an irreducible representation of $\mathfrak {sl}_{2}$ with the highest weight $\varsigma \leq 2$ , then for any $v=b_{0}v_{0}+\cdots +b_{\varsigma }v_{\varsigma }\in V^{\varsigma }$ , we have

(2.5) $$ \begin{align} \exp(tU).v=&\sum_{j=0}^{\varsigma}\sum_{i=0}^{j}b_{i}\binom{j}{i}t^{j-i}v_{j}, \; \end{align} $$

(2.6) $$ \begin{align} \hphantom{\hspace{-17pt}}\exp(\omega Y_{n}).v=& \sum_{j=0}^{\varsigma} b_{j}e^{ (\varsigma-2j)\omega /2} v_{j}. \; \end{align} $$

For elements $g\in \exp \mathfrak {g}$ close to identity, we decompose

$$ \begin{align*}g=h\exp(v),\quad h\in \mathrm{SO}_{0}(2,1),\quad v\in V^{\perp}\end{align*} $$

where $\mathrm {SO}_{0}(2,1)$ is the connected component of $\mathrm {SO}(2,1)$ . Moreover, it is convenient to think about $h\in \mathrm {SO}_{0}(2,1)$ as a $(2\times 2)$ -matrix with determinant $1$ . Thus, consider the two-to-one isogeny $\iota :SL_{2}(\mathbb {R})\rightarrow \mathrm {SO}(2,1)\subset G$ induced by ${\mathfrak {sl}_{2}(\mathbb {R})\rightarrow \operatorname {\mathrm {Span}}\{U,Y_{n},\bar {U}\} \subset \mathfrak {g}}$ . In the following, for $h\in \mathrm {SO}_{0}(2,1)$ and v in an irreducible representation, we write

$$ \begin{align*}h=\left[ \begin{array}{ccc} a & b \\ c & d\\ \end{array} \right]\!,\quad v=b_{0}v_{0}+\cdots+b_{\varsigma}v_{\varsigma}\end{align*} $$

where $v_{i}$ are weight vectors in $\mathfrak {g}$ of weight i. Notice that h should more appropriately be written as $\iota (h)$ . In addition, for notational simplicity, we shall usually assume that ${v\in V^{\perp }}$ lies in a single irreducible representation, since the proofs will mostly focus on the $\operatorname {\mathrm {Ad}} u^{t}$ -action and so the general case will be identical but tedious to write down.

For the centralizer $ C_{\mathfrak {g}}(U)$ (for an arbitrary Lie algebra $\mathfrak {g}$ ), we have the corresponding decomposition:

(2.7) $$ \begin{align} C_{\mathfrak{g}}(U)=\mathbb{R}U\oplus V^{\perp}_{C}, \end{align} $$

where $V^{\perp }_{C}\subset V^{\perp }$ consists of the highest weight vectors other than U. In particular, for $\mathfrak {g}=\mathfrak {so}(n,1)$ , under the setting of equation (2.2), one may calculate

(2.8) $$ \begin{align} C_{\mathfrak{g}}(U)&=\mathbb{R}U\oplus V^{\perp}_{C}=\mathbb{R}U\oplus \mathfrak{k}^{\perp}_{C}\oplus \mathfrak{n}^{\perp}_{C} \;\nonumber\\ &= \mathbb{R}U\oplus\left[ \begin{array}{ccc} \mathfrak{so}(n-2) & \\ & 0 \\ \end{array} \right] \oplus\left\{ \left[ \begin{array}{cccc} 0& 0 & \mathbf{u} & \mathbf{u} \\ 0& 0 & 0 & 0 \\ -\mathbf{u}^{T} & 0 & 0 & 0\\ \mathbf{u}^{T} &0 &0 & 0 \\ \end{array} \right]: \mathbf{u}\in\mathbb{R}^{n-2}\right\}\!. \; \end{align} $$

Note that $\mathfrak {k}^{\perp }_{C}$ consists of semisimple elements and $\mathfrak {n}^{\perp }_{C}$ consists of nilpotent elements, and they satisfy $[\mathfrak {k}^{\perp }_{C},\mathfrak {n}^{\perp }_{C}]=\mathfrak {n}^{\perp }_{C}$ .

2.3 Time changes

Let Y be a homogeneous space and U be a nilpotent. Let $\phi ^{U,\tau }_{t}$ be a time change for the unipotent flow $\phi ^{U}_{t}$ , $t\in \mathbb {R}$ . Thus, we assume that:

• • $\tau :Y\rightarrow \mathbb {R}^{+}$ is a integrable non-negative function on Y satisfying

  $$ \begin{align*}\int_{Y}\tau(y)\,dm_{Y}(y)=1; \end{align*} $$

• • $\xi :Y\times \mathbb {R}\rightarrow \mathbb {R}$ is the cocycle determined by

  $$ \begin{align*} t=\int_{0}^{\xi(y,t)}\tau(u^{s}y)\,ds=\int_{0}^{\xi(y,t)}\tau(\phi_{t}^{U} y)\,ds; \end{align*} $$

• • $\phi ^{U,\tau }_{t}:Y\rightarrow Y$ is given by the relation

  

In practice, we define $z:Y\times \mathbb {R}\rightarrow \mathbb {R}$ by

$$ \begin{align*}z(y,t)=\int_{0}^{t}\tau(u^{s}y)\,ds.\end{align*} $$

It follows that

(2.9) $$ \begin{align} t=z(y,\xi(y,t)),\quad \phi^{U,\tau}_{z(y,t)}(x)=\phi^{U}_{t}(y)= u^{t}y. \end{align} $$

Let $\kappa>0$ and $\mathbf {K}_{\kappa }(Y)$ be the collection of all positive integrable functions $\tau $ on Y such that $\tau ,\tau ^{-1}$ are bounded and satisfy

(2.10) $$ \begin{align} \bigg|\int_{Y}\tau(y)\tau(u^{t}y)\,d\nu(y)-\bigg(\int_{Y}\tau(y)\nu(y)\bigg)^{2}\bigg|\leq D_{\tau}|t|^{-\kappa} \end{align} $$

for some $D_{\tau }>0$ . Let $\mathbf {K}(Y)=\bigcup _{\kappa>0}\mathbf {K}_{\kappa }(Y)$ . This is the effective mixing property of the unipotent flow $\phi _{t}^{U}$ . Note that [Reference Kleinbock and MargulisKM99] (see also [Reference VenkateshVen10]) has shown that there is $\kappa>0$ such that

$$ \begin{align*}\bigg|\langle\phi^{U}_{t}(f),g\rangle-\bigg(\int_{Y}f(y)\nu(y)\bigg)\bigg(\int_{Y}g(y)\nu(y)\bigg)\bigg|\ll(1+|t|)^{-\kappa}\|f\|_{W^{s}}\|g\|_{W^{s}}\end{align*} $$

for $f,g\in C^{\infty }(X)$ , where $s\geq \dim (K)$ and $W^{s}$ denotes the Sobolev norm on $Y=G/\Gamma $ . According to Lemma 3.1 [Reference RatnerRat86], when $\tau \in \mathbf {K}_{\kappa }(Y)$ , we have the effective ergodicity: there is $K\subset Y$ with $\nu (K)>1-\sigma $ and $t_{K}>0$ such that

(2.11) $$ \begin{align} |t-z(y,t)|=O(t^{1-\kappa}) \end{align} $$

for all $t\geq t_{K}$ and $y\in K$ . Later on, we shall make use of the effective mixing/ergodicity to study the shearing property of unipotent flows (see §4 and equation (5.1)).

2.4 Cohomology

We first introduce the $1$ -coboundary of two functions.

Definition 2.1. (Cohomology)

We say that two functions $\tau _{1},\tau _{2}$ on Y are measurable (respectively $L^{2}$ , smooth, etc.) cohomologous over the flow $\phi _{t}$ if there exists a measurable (respectively $L^{2}$ , smooth, etc.) function f on Y, called the transfer function, such that

(2.12) $$ \begin{align} \int_{0}^{T}\tau_{1}(\phi_{t}y)-\tau_{2}(\phi_{t}y)\,dt=f(\phi_{T}y)-f(y). \end{align} $$

For $i\in \{1,2\}$ , let $(Y_{i},\mathcal {Y}_{i},\nu _{i},\phi _{t}^{(i)})$ be measure-preserving flows, and let $\tau _{i}:Y_{i}\rightarrow \mathbb {R}$ be measurable functions on $Y_{i}$ . In addition, we extend $\tau _{i}$ to $Y_{1}\times Y_{2}$ by setting

$$ \begin{align*}\tau_{i}: (y_{1},y_{2})\mapsto\tau_{i}(y_{i}),\quad i=1,2. \end{align*} $$

Definition 2.2. (Joint cohomology)

Let $\rho \in J(\phi _{t}^{(1)},\phi _{t}^{(2)})$ be a joining of $\phi _{t}^{(1)}$ and $\phi _{t}^{(2)}$ . We say that $\tau _{1}$ and $\tau _{2}$ are jointly cohomologous via $\rho $ if $\tau _{1}$ and $\tau _{2}$ (considered as functions on $Y_{1}\times Y_{2}$ ) are cohomologous over $\phi _{t}^{(1)}\times \phi _{t}^{(2)}$ on $(Y_{1}\times Y_{2},\rho )$ . More specifically, if $\tau _{1}$ and $\tau _{2}$ are cohomologous over $\phi _{t}^{(1)}\times \phi _{t}^{(2)}$ with a transfer function $f:Y_{1}\times Y_{2}\rightarrow \mathbb {R}$ , then we say that $\tau _{1}$ and $\tau _{2}$ are jointly cohomologous via $(\rho ,f)$ , and we have

(2.13) $$ \begin{align} \int_{0}^{T}(\tau_{1}-\tau_{2})(\phi^{(1)}_{t}y_{1}, \phi^{(2)}_{t}y_{2})\,dt= f(\phi^{(1)}_{T}y_{1}, \phi^{(2)}_{T}y_{2})-f(y_{1}, y_{2}) \end{align} $$

for $\rho $ -a.e. $(y_{1},y_{2})\in Y_{1}\times Y_{2}$ and all $T\in \mathbb {R}$ .

Let , . Then there is a unique family $\{\rho _{y_{1}}^{\mathcal {A}_{1}}:y_{1}\in Y_{1}\}$ of probability measure, called the conditional measures, on $Y_{2}$ such that

(2.14) $$ \begin{align} E^{\rho}(g|\mathcal{A}_{1})(y_{1})=\int_{Y_{2}}g(y_{1},y_{2})d\rho_{y_{1}}^{\mathcal{A}_{1}}(y_{2}),\quad \rho^{\mathcal{A}_{1}}_{\phi^{(1)}_{t}y_{1}}=(\phi^{(2)}_{t})_{\ast}\rho^{\mathcal{A}_{1}}_{y_{1}} \end{align} $$

for every $g\in L^{1}(Y_{1}\times Y_{2},\rho )$ , $t\in \mathbb {R}$ , and $\nu _{1}$ -a.e. $y_{1}\in Y_{1}$ . Taking the integration over $\rho _{y_{1}}^{\mathcal {A}_{1}}$ , the expressions of equations (2.13) and (2.14) show that if the transfer function $f(y_{1},\cdot )\in L^{1}(Y_{2},\rho ^{\mathcal {A}_{1}}_{y_{1}})$ for $\nu _{1}$ -a.e. $y_{1}\in Y_{1}$ , then $\tau _{1}$ and $E^{\rho }(\tau _{2}|\mathcal {A}_{1})$ are cohomologous along $\phi ^{(1)}_{t}$ via $E^{\rho }(f|\mathcal {A}_{1})$ . We have just proved the following proposition.

3.1 Joinings

Let $G=\mathrm {SO}(n,1)$ , $\Gamma $ be a lattice of G, $(X,\mu )$ be the homogeneous space $X=G/\Gamma $ equipped with the Lebesgue measure $\mu $ , and let $\phi ^{U}_{t}$ be a unipotent flow on X. Let $(Y,\nu ,S)$ be a measure-preserving system. We want to study the joinings of $(X,\mu ,\phi _{1}^{U})$ and $(Y,\nu ,S)$ . Thus, let $\rho $ be an ergodic joining of $\phi _{1}^{U}$ and S, that is, $\rho $ is a probability measure on $X\times Y$ , whose marginals on X and Y are $\mu $ and $\nu $ , respectively, and which is $(\phi _{1}^{U}\times S)$ -ergodic.

Let $C(\phi _{1}^{U})$ be the commutant of $\phi _{1}^{U}$ , that is, the collection of all measure-preserving transformations on X that commute with $\phi _{1}^{U}$ . The following is a basic criterion for $\rho $ in terms of the commutant of $\phi _{1}^{U}$ .

Lemma 3.1. Let the notation and assumptions be as above. Assume further that $T\in C(\phi _{1}^{U})$ is ergodic on $(X,\mu )$ . Then

$$ \begin{align*} \text{either }(T\times\operatorname{\mathrm{id}})_{\ast}\rho\perp\rho\quad \text{or}\quad\rho=\mu\times\nu.\end{align*} $$

Proof. First, by the commutative property of T, we easily see that $(T\times \operatorname {\mathrm {id}})_{\ast }\rho $ is again $(\phi _{1}^{U}\times S)$ -ergodic on $X\times Y$ . It implies that either $(T\times \operatorname {\mathrm {id}})_{\ast }\rho \perp \rho $ or $(T\times \operatorname {\mathrm {id}})_{\ast }\rho =\rho $ . Now assume that $(T\times \operatorname {\mathrm {id}})_{\ast }\rho =\rho $ , that is, $\rho $ is $(T\times \operatorname {\mathrm {id}})$ -invariant. Then via disintegration, we know that $\rho _{y}$ is T-invariant on X for $\nu $ -a.e. $y\in Y$ , where

(3.1) $$ \begin{align} \rho=\int_{Y}\rho_{y}\,d\nu(y). \end{align} $$

Now assume for contradiction that there exists $B\subset Y$ with $\nu (B)>0$ such that $\rho _{y}\neq \mu $ for $y\in B$ . It follows that for $y\in B$ , there is $A_{y}\subset X$ with $\mu (A_{y})>0$ such that, for $x\in A_{y}$ , we have

(3.2) $$ \begin{align} (\rho_{y})_{x}^{\mathcal{E}}\neq\mu, \end{align} $$

where $(\rho _{y})_{x}^{\mathcal {E}}$ is given by the T-ergodic decomposition

$$ \begin{align*}\rho_{y}=\int_{X}(\rho_{y})_{x}^{\mathcal{E}}\,d\mu(x).\end{align*} $$

Notice that by the ergodicity, there is a $\mu $ -conull set $\Omega \subset X$ , namely the set of T-generic points of $\mu $ , such that $(\rho _{y})_{x}^{\mathcal {E}}(\Omega )=0$ for the measures $(\rho _{y})_{x}^{\mathcal {E}}$ in equation (3.2). Then by the assumption of joining, we have

$$ \begin{align*} \mu(\Omega)=\rho(\pi_{X}^{-1}(\Omega))&= \int_{Y}\rho_{y} (\Omega)\,d\nu(y)\;\nonumber\\ &= \int_{B}\rho_{y}(\Omega)\,d\nu(y)+\int_{Y\setminus B}\rho_{y}(\Omega)\,d\nu(y)\;\nonumber\\ &\leq \int_{B}\int_{X}(\rho_{y})_{x}^{\mathcal{E}}(\Omega)\,d\mu(x)\,d\nu(y)+ \nu(Y\setminus B)\;\nonumber\\ &= \int_{B}\int_{X\setminus A_{y}}(\rho_{y})_{x}^{\mathcal{E}}(\Omega)\,d\mu(x)\,d\nu(y)+ \nu(Y\setminus B)\;\nonumber\\ &\leq \int_{B} \mu(X\setminus A_{y})\,d\nu(y)+ \nu(Y\setminus B)\;\nonumber\\ &<\nu(B)+\nu(Y\setminus B)=1,\nonumber \end{align*} $$

which is a contradiction. Thus, we conclude that $\rho _{y}=\mu $ for $\nu $ -a.e. $y\in Y$ and so $\rho = \mu \times \nu $ .

By Moore’s ergodicity theorem, we deduce the following corollary.

Corollary 3.2. If $w\in C_{\mathfrak {g}}(U)$ so that $\langle \exp tw\rangle _{t\in \mathbb {R}}$ is not compact, then

$$ \begin{align*}\text{either }(\phi^{w}_{1}\times\operatorname{\mathrm{id}})_{\ast}\rho\perp\rho\quad\text{or}\quad \rho=\mu\times\nu.\end{align*} $$

3.2 H-property

In this section, we want to introduce the H-property (or Ratner property) to study the joining $\rho $ in terms of the unipotent flow $\phi ^{U}_{t}$ on X. The classic H-property can be formulated as the following theorem.

Remark 3.4. In fact, for $x_{2}=gx_{1}$ with $g=\exp (v)\in B^{G}_{\delta }$ , the element $q\in C_{\mathfrak {g}}(U)$ in Theorem 3.3 is chosen by

(3.4) $$ \begin{align} q=\pi_{C_{\mathfrak{g}}(U)}\exp(LU).v, \end{align} $$

where $\pi _{C_{\mathfrak {g}}(U)}: \mathfrak {g}\rightarrow C_{\mathfrak {g}}(U)$ is the natural projection and $\exp (LU).v$ is the adjoint representation (see equation (2.5)). We often call q as the fastest relative motion between $x_{1},x_{2}$ ; see [Reference MorrisMor05] for more discussion. In what follows, we choose $Q=B_{\unicode{x3bb} }^{C_{G}(u)}$ to be the ball of radius $\unicode{x3bb} $ of e in $C_{G}(u)$ for sufficiently small $\unicode{x3bb} $ (independent of $\epsilon $ ), and then $\partial Q$ is the sphere of radius $\unicode{x3bb} $ . Now by equations (2.3) and (2.4), we have the decomposition

$$ \begin{align*} v=v_{0}+v_{2}, \end{align*} $$

where $v_{0}\in \sum _{i} V_{i}^{0}$ and $v_{2}\in \mathfrak {sl}_{2}+\sum _{j} V_{j}^{2}$ . Thus, $\|v_{0}\|,\|v_{2}\|<\delta $ and

$$ \begin{align*} q=v_{0}+\pi_{C_{\mathfrak{g}}(U)}\exp(LU).v_{2}. \end{align*} $$

Since $\|q\|=\unicode{x3bb} $ , we see that $v_{0}$ is negligible. In other words, we can replace q by

(3.5)

and then Theorem 3.3 still holds. However, note that $q^{\prime }\in \mathfrak {n}= \mathbb {R}U+\mathfrak {n}_{C}^{\perp }$ (cf. equation (2.8)). Thus, the one-parameter group $\langle \exp (tq^{\prime })\rangle _{t\in \mathbb {R}}$ generated by $q^{\prime }$ is not compact.

In the following, we shall generalize the idea in [Reference RatnerRat83] and prove Theorem 1.1.

Theorem 3.5. Let the notation and assumptions be as above. Then either $\rho =\mu \times \nu $ or $(\phi ^{U}_{1}\times S,\rho )$ is a compact extension of $(S,\nu )$ . More precisely, if $\rho \neq \mu \times \nu $ , then there exists a $\nu $ -conull set $\Theta \subset Y$ , a compact subgroup $C^{\rho }\subset C_{G}(u)$ , and $n>0$ such that, for any $y\in \Theta $ , there exist $x_{1}^{y},\ldots ,x_{n}^{y}$ in the support of $\rho _{y}$ with

$$ \begin{align*}\rho_{y}(C^{\rho}x_{i}^{y})=\frac{1}{n}\end{align*} $$

for $i=1,\ldots ,n$ , where $ \rho =\int _{Y}\rho _{y}\,d\nu (y)$ is the disintegration along Y (cf. equation (3.1)).

Assume that $\rho \neq \mu \times \nu $ . Then by Corollary 3.2, there is a $\rho $ -conull set $\Omega \subset X\times Y$ , namely the set of $(\phi _{1}^{U}\times S)$ -generic points, such that $(\phi ^{w}_{1}\times \operatorname {\mathrm {id}}) (\Omega )\cap \Omega =\emptyset $ for all $w\in \mathfrak {n}$ . Given a sufficiently small $\unicode{x3bb}>0$ , we define the sphere of radius $\unicode{x3bb} $ of $0$ by

Then, one can find a compact subset $K_{1}\subset \Omega $ with $\mu (K_{1})>199/200$ . Then,

$$ \begin{align*} \bigcup_{w\in B^{\mathfrak{n}}_{\unicode{x3bb}}}(\phi^{w}_{1}\times \operatorname{\mathrm{id}})(K_{1}) \end{align*} $$

is compact. Thus, there are $\epsilon>0$ and $K_{2}\subset K_{1}$ with $\mu (K_{2})>99/100$ such that

$$ \begin{align*}d_{X\times Y}\bigg(K_{2},\bigcup_{w\in B^{\mathfrak{n}}_{\unicode{x3bb}}}(\phi^{w}_{1}\times \operatorname{\mathrm{id}})(K_{1}) \bigg)>\epsilon. \end{align*} $$

It follows that if $(x_{1},y),(x_{2},y)\in K_{2}$ then

(3.6) $$ \begin{align} d_{X}(x_{2},\phi^{w}_{1}x_{1})\geq\epsilon \end{align} $$

for all $w\in B^{\mathfrak {n}}_{\unicode{x3bb} }$ . Let $\alpha =\alpha (\epsilon )>0$ be as in Theorem 3.3. Comparing equation (3.6) with equation (3.3), we conclude the following lemma.

Proof. Suppose that M, $\delta $ , $K_{4}$ are given, and $x_{2}\not \in C_{G}(u)x_{1}$ with $d_{X}(x_{1},x_{2})<\delta $ . Then by the H-property of the unipotent flow (Theorem 3.3 and Remark 3.4), we know that there are $L>M/\alpha $ and $w\in B^{\mathfrak {n}}_{\unicode{x3bb} }$ such that

(3.7) $$ \begin{align} d_{X}(\phi^{U}_{n}x_{1},\phi^{w}_{1}\phi^{U}_{n}x_{2})<\epsilon \end{align} $$

for $n\in [L,(1+\alpha )L]$ . Next, we shall find some qualified $x_{1},x_{2}\in X$ such that the distance between $\phi ^{U}_{n}x_{1}$ and $\phi ^{w}_{1}\phi ^{U}_{n}x_{2}$ is at least $\epsilon $ . This will lead to a contradiction.

First, applying the ergodic theorem, there is a measurable set $K_{3}\subset \Omega $ with $\rho (K_{3})>1-\alpha /2(100+\alpha )$ , a number $M_{1}>0$ such that

(3.8) $$ \begin{align} \frac{1}{n}|\{k\in [0,n]:(\phi^{U}_{1}\times S)^{k}(x,y)\in K_{2}\}|> \frac{9}{10} \end{align} $$

for $(x,y)\in K_{3}$ and $n>M_{1}$ . Applying the ergodic theorem one more time, there is a measurable set $K_{4}\subset \Omega $ with $\rho (K_{4})>0$ , a number $M_{2}>0$ such that

(3.9) $$ \begin{align} \frac{1}{n}|\{k\in [0,n]:(\phi^{U}_{1}\times S)^{k}(x,y)\in K_{3}\}|> 1-\frac{\alpha}{10+\alpha} \end{align} $$

for $(x,y)\in K_{4}$ and $n>M_{2}$ .

Choose $M=\max \{M_{1},M_{2}\}$ and then $L>M/\alpha $ and $\delta =\delta (\epsilon ,M)>0$ as obtained from the H-property (Theorem 3.3). Let $(x_{1},y),(x_{2},y)\in K_{4}$ with $d_{X}(x_{1},x_{2})<\delta $ . Then replacing n by $(1+\alpha /10)L$ and applying equation (3.9), we know that

$$ \begin{align*}(\phi^{U}_{1}\times S)^{s}(x_{1},y),(\phi^{U}_{1}\times S)^{t}(x_{2},y)\in K_{3}\end{align*} $$

for some integers $s,t\in [L,(1+\alpha /10)L]$ . Further, replacing the interval $[0,n]$ by $[s,(1+\alpha )L]$ (respectively $[t,(1+\alpha )L]$ ) and applying equation (3.8), we know that

$$ \begin{align*}\frac{1}{(1+\alpha)L-s}|\{k\in [s,(1+\alpha)L]:(\phi^{U}_{1}\times S)^{k}(x_{1},y)\in K_{2}\}|> \frac{9}{10}, \end{align*} $$

$$ \begin{align*}\frac{1}{(1+\alpha)L-t}|\{k\in [t,(1+\alpha)L]:(\phi^{U}_{1}\times S)^{k}(x_{2},y)\in K_{2}\}|> \frac{9}{10}. \end{align*} $$

It follows that there exists $n\in [(1+\alpha /10)L,(1+\alpha )L]$ such that

$$ \begin{align*}(\phi^{U}_{1}\times S)^{n}(x_{1},y),\ (\phi^{U}_{1}\times S)^{n}(x_{2},y)\in K_{2}.\end{align*} $$

Then by equation (3.6), we have

$$ \begin{align*} d_{X}(\phi^{U}_{n}x_{1},\phi^{w}_{1}\phi^{U}_{n}x_{2})\geq\epsilon, \end{align*} $$

which contradicts equation (3.7).

Recall that via disintegration (cf. equation (3.1)), we have

$$ \begin{align*} \rho=\int_{Y}\rho_{y}\,d\nu(y). \end{align*} $$

Then by the ergodic theory, we have the following lemma.

Lemma 3.7. Assume that $\rho \neq \mu \times \nu $ . There exists a $\nu $ -conull set $\Theta \subset Y$ and $n>0$ such that, for any $y\in \Theta $ , there exist $x_{1}^{y},\ldots ,x_{n}^{y}$ in the support of $\rho _{y}$ with

$$ \begin{align*}\rho_{y}(C_{G}(u)x_{i}^{y})=\frac{1}{n} \end{align*} $$

for $i=1,\ldots ,n$ .

Proof. Let $f:Y\rightarrow \mathbb {R}$ be defined by

$$ \begin{align*} f:y\mapsto\sup_{x\in X}\rho_{y}(C_{G}(u)x). \end{align*} $$

By Lemma 3.6, we know that for , $f(y)>0$ . Note also that $\nu (K_{4}^{Y})>0$ and f is S-invariant. By the ergodicity, f is a positive constant, say $f\equiv c$ , on a $\nu $ -conull set $\Theta _{1}\subset Y$ .

Next, consider

Then D is $(\phi ^{U}_{1}\times S)$ -invariant and $\rho (D)>0$ . Thus, $\rho (D)=1$ . Next, define

Then $\Theta \subset Y$ is an S-invariant $\nu $ -conull set. Thus, for any $y\in \Theta $ , we have

$$ \begin{align*}\rho_{y}(C_{G}(u)x)\equiv c \end{align*} $$

for any $x\in X$ with $(x,y)\in D$ . It forces $n=1/c$ to be an integer. In addition, for any $y\in \Theta $ , there are only finitely many points $x_{1}^{y},\ldots ,x_{n}^{y}$ with

$$ \begin{align*} \rho_{y}(C_{G}(u)x_{i}^{y})=\frac{1}{n} \end{align*} $$

for $i=1,\ldots ,n$ .

Thus, by Lemma 3.7, we see that $\rho _{y}$ supports on $\bigsqcup _{i=1}^{n}C_{G}(u)x_{i}^{y}$ whenever $y\in \Theta $ . With a further effort, we observe that these $\rho _{y}$ must have a compact support.

Proof of Theorem 3.5

For a Borel measurable subset $A\subset C_{G}(u)$ , consider the map $f_{A}:X\times Y\rightarrow \mathbb {R}^{+}$ be defined by

$$ \begin{align*} f_{A}:(x,y)\mapsto \rho_{y}(Ax). \end{align*} $$

Note that since $\rho $ is $(\phi ^{U}_{1}\times S)$ -invariant, we have

$$ \begin{align*} (\phi^{U}_{1})_{\ast}\rho_{y}=\rho_{Sy}. \end{align*} $$

It follows that

$$ \begin{align*}f_{A}(x,y)=\rho_{y}(Ax)=\rho_{Sy}(\phi^{U}_{1}Ax)=\rho_{Sy}(A\phi^{U}_{1}x)=f_{A}(\phi^{U}_{1}x,Sy).\end{align*} $$

In other words, $f_{A}$ is $(\phi ^{U}_{1}\times S)$ -invariant and therefore is $\rho $ -almost everywhere a constant, say $m(A)$ . Thus, for any $A\in \mathcal {B}(C_{G}(u))$ , there exists a $\rho $ -conull set $\Omega _{A}\subset X\times Y$ , such that

(3.10) $$ \begin{align} \rho_{y}(Ax)\equiv m(A) \end{align} $$

for $(x,y)\in \Omega _{A}$ .

Next, we consider the fundamental domain, that is, a Borel subset $F\subset C_{G}(u)$ such that the natural map $F\rightarrow C_{G}(u)/(C_{G}(u)\cap \Gamma )$ defined by $g\mapsto g\Gamma $ is bijective. Then since $\mathcal {B}(F)$ is countably generated, by Carathéodory’s extension theorem, we know that $m:\mathcal {B}(F)\rightarrow \mathbb {R}^{+}$ is a measure. In addition, it follows from equation (3.10) that there exists a $\rho $ -conull set $\Omega \subset X\times Y$ , such that

(3.11) $$ \begin{align} \rho_{y}(Ax)\equiv m(A) \end{align} $$

for $(x,y)\in \Omega $ , $A\in \mathcal {B}(F)$ .

Now assume that equation (3.11) holds for $(x,y),(gx,y)\in \Omega $ and $g\in C_{G}(u)$ . Then,

$$ \begin{align*}m(A)=\rho_{y}(Agx)=m(Ag) \end{align*} $$

for $A\in \mathcal {B}(F)$ . In other words, m is g-(right) invariant and so is (right) Haar. Note that $C_{G}(u)$ is unimodular (since its Lie algebra $C_{\mathfrak {g}}(U)$ is a direct sum of a compact and a nilpotent Lie subalgebra). We conclude that m is also a (left) Haar measure, and therefore $\rho _{y}$ is (left) Haar on $C_{G}(u)x$ for $(x,y)\in \Omega $ .

Let $C^{\rho }$ be the stabilizer of m. Then the above result shows that $\rho $ is $(C^{\rho }\times \operatorname {\mathrm {id}})$ -invariant. Thus, according to Corollary 3.2, $C^{\rho }$ must be compact. This finishes the proof of Theorem 3.5.

Using Theorem 3.5, for any ergodic joining $\rho $ of $\phi _{1}^{U}$ and S on $X\times Y$ , we obtain an ergodic joining of $\phi _{1}^{U}$ and S on $C^{\rho }\backslash X\times Y$ under the natural projection $\pi : X\times Y\rightarrow C^{\rho }\backslash X\times Y$ . Moreover, when $\overline {\rho }\neq \overline {\mu }\times \nu $ is not the product measure, it is a finite extension of $\nu $ , that is, $\operatorname {\mathrm {supp}}\overline {\rho }_{y}$ consists of exactly n points $\overline {x}_{1}^{y},\ldots ,\overline {x}_{n}^{y}$ for $\nu $ -a.e. $y\in Y$ (without loss of generality, we shall assume that it holds for all $y\in Y$ ). Note that $y\mapsto \overline {x}_{i}^{y}$ need not be measurable. However, this can be resolved by using Kunugui’s theorem (see [Reference KallmanKal75, Reference KunuguiKun40]).

Therefore, let , $\pi _{X}:X\times Y\rightarrow X$ , $\pi _{\overline {X}}:\overline {X}\times Y\rightarrow \overline {X}$ , $\pi _{Y}:\overline {X}\times Y\rightarrow Y$ be the natural projections. By Kunugui’s theorem, we are able to find $\hat {\psi }_{i}:Y\rightarrow \overline {X}\times Y$ for $i=1,\ldots ,n$ such that $\pi _{Y}\circ \hat {\psi }_{i}=\operatorname {\mathrm {id}}$ and $\hat {\psi }_{i}(Y)\cap \hat {\psi }_{j}(Y)=\emptyset $ whenever $i\neq j$ . Let

(3.12)

Then $\rho (\Omega _{i})=1/n$ , $ \bigcup \Omega _{i}=\operatorname {\mathrm {supp}}\overline {\rho }$ , and $\Omega \cap \operatorname {\mathrm {supp}}\overline {\rho }_{y}$ consists of exactly one point. Next, we can apply Kunugui’s theorem again and obtain $\psi _{i}:Y\rightarrow X$ so that $P_{X}\circ \psi _{i}=\overline {\psi }_{i}$ , where $P_{X}:X\rightarrow \overline {X}$ .

4.1 Preliminaries

We start with a combinatorial result. Let I be an interval in $\mathbb {R}$ and let $J_{i},J_{j}$ be disjoint subintervals of $I,J_{i}=[x_{i},y_{i}]$ , $y_{i}<x_{j}$ if $i<j$ . Denote

For a collection $\beta $ of finitely many intervals, we define

In addition, for a collection $\beta $ of finitely many intervals, an interval I, let

Proposition 4.1. (Existence of large intervals, Solovay [Reference RatnerRat79])

Given $\eta \in (0,1)$ , $\zeta \in (0,1)$ , there is $\theta =\theta (\zeta ,\eta )\in (0,1)$ such that if I is an interval of length $\unicode{x3bb} \gg 1 $ and $\alpha =\{J_{1},\ldots ,J_{n}\}=\mathcal {G}\cup \mathcal {B}$ is a partition of I into good and bad intervals such that:

1. (1) for any two good intervals $J_{i},J_{j}\in \mathcal {G}$ , we have

   (4.1) $$ \begin{align} d(J_{i},J_{j})\geq[\min\{\operatorname{\mathrm{Leb}}(J_{i}),\operatorname{\mathrm{Leb}}(J_{j})\}]^{1+\eta}; \end{align} $$

2. (2) $ \operatorname {\mathrm {Leb}}(J)\leq \zeta \unicode{x3bb} $ for any good interval $J\in \mathcal {G}$ ;

3. (3) $ \operatorname {\mathrm {Leb}}(J)\geq 1$ for any bad interval $J\in \mathcal {B}$ ;

then the measure of bad intervals $\operatorname {\mathrm {Leb}}(\bigcup _{J\in \mathcal {B}}J)\geq \theta \unicode{x3bb} $ . More precisely, we can take

$$ \begin{align*} \theta=\theta(\zeta,\eta)= \prod_{n=0}^{\infty} (1+ C\zeta^{n\eta})^{-1} \end{align*} $$

for some constant $C>0$ (independent of $\zeta ,\eta $ ).

Proof. Assume that $\zeta ^{1-k} \leq \unicode{x3bb} \leq \zeta ^{-k}$ for some $k\geq 1$ . Let

, and $\mathcal {B}_{\leq n}$ be the collection of the remaining intervals forming $I\setminus \bigcup _{J\in \mathcal {G}_{\leq n}}J$ . Then for $n\in \mathbb {N}$ , $J\in \mathcal {B}_{\leq n}$ , by equation (4.1), we have

$$ \begin{align*} \frac{|\mathcal{B}_{\leq n+1}\cap J|}{\operatorname{\mathrm{Leb}}(J)}&=\frac{|\mathcal{B}_{\leq n+1}\cap J|}{|\mathcal{G}_{n+1}\cap J|+|\mathcal{B}_{\leq n+1}\cap J|}= \bigg(1+\frac{|\mathcal{G}_{n+1}\cap J|}{|\mathcal{B}_{\leq n+1}\cap J|}\bigg)^{-1} \;\nonumber\\ &\geq \bigg(1+\frac{l\zeta^{n+1}\unicode{x3bb}}{(l-1) \zeta^{(n+2)(1+\eta)}\unicode{x3bb}^{1+\eta}}\bigg)^{-1} = (1+ C\zeta^{(k-n)\eta})^{-1}, \; \nonumber \end{align*} $$

where $l\geq 2$ is the number of intervals in $\mathcal {G}_{n+1}\cap J$ , and $C>0$ is some constant depending on $\eta $ and $\zeta $ . One can also show that when $k=0,1$ , we have a similar relation. By summing over $J\in \mathcal {B}_{\leq n}$ , we obtain

$$ \begin{align*}\frac{|\mathcal{B}_{\leq n+1}|}{|\mathcal{B}_{\leq n}|}\geq(1+ C\zeta^{(k-n)\eta})^{-1}. \end{align*} $$

Note that by item (2), $|\mathcal {B}_{\leq 0}|=\unicode{x3bb} $ , and by item (3), $\mathcal {B}_{\leq n}=\mathcal {B}_{\leq n+1}$ for all $n\geq k$ . We calculate

$$ \begin{align*}|\mathcal{B}|=|\bigcap_{k\geq 0}\mathcal{B}_{\leq k}|=\lim_{k\rightarrow\infty}|\mathcal{B}_{\leq k}|=\prod_{n=0}^{\infty}\frac{|\mathcal{B}_{\leq n+1}|}{|\mathcal{B}_{\leq n}|}\cdot \unicode{x3bb}\geq \prod_{n=0}^{k} (1+ C\zeta^{(k-n)\eta})^{-1}\cdot \unicode{x3bb}.\end{align*} $$

Now note that

$$ \begin{align*}\theta(\zeta,\eta)=\prod_{n=0}^{\infty} (1+ C\zeta^{n\eta})^{-1}\leq\prod_{n=0}^{k} (1+ C\zeta^{(k-n)\eta})^{-1} \end{align*} $$

and the proposition follows.

In light of equation (4.1), we make the following definition.

Definition 4.1. (Effective gaps between intervals)

We say that two intervals $I,J\subset \mathbb {R}$ have an effective gap if

$$ \begin{align*} d(I,J)\geq[\min\{\operatorname{\mathrm{Leb}}(I),\operatorname{\mathrm{Leb}}(J)\}]^{1+\eta} \end{align*} $$

for some $\eta>0$ . Later, we shall obtain some quantitative results relative to the effective gap.

In the following, we shall use the asymptotic notation:

• • $A\ll B$ or $A=O(B)$ means there is a constant $C>0$ such that $A\leq CB$ (we also write $A\ll _{\kappa } B$ if the constant $C(\kappa )$ depends on some coefficient $\kappa $ );

• • $A=o(B)$ means that $A/B\rightarrow 0$ as $B\rightarrow 0$ ;

• • $A\asymp B$ means there is a constant $C>1$ such that $C^{-1}B\leq A\leq CB$ ;

• • $A\approx 0$ means $A\in (0,1)$ close to $0$ , and $A\approx 1$ means $A\in (0,1)$ close to $1$ .

Similar to [Reference TangTan22], we need to following quantitative property of polynomials.

Lemma 4.3. Fix numbers $R_{0}>0, \kappa \in (0,1]$ , a real polynomial $p(x)=v_{0}+v_{1}x+\cdots +v_{k}x^{k}\in \mathbb {R}[x]$ . Assume further that there exist intervals $[0,\overline {l}_{1}]\cup [l_{2},\overline {l}_{2}]\cup \cdots \cup [l_{m},\overline {l}_{m}]$ such that

(4.2) $$ \begin{align} |p(t)|\ll\max\{R_{0},t^{1-\kappa}\} \quad\text{if and only if}\, t\in[0,\overline{l}_{1}]\cup[l_{2},\overline{l}_{2}]\cup\cdots\cup[l_{m},\overline{l}_{m}]. \end{align} $$

Then $\overline {l}_{1}$ has the lower bound l depending on $\max _{i}|v_{i}|$ , $R_{0}$ , $\kappa $ , and the implicit constant such that $l\nearrow \infty $ as $\max _{i}|v_{i}|\searrow 0$ for fixed $R_{0},\kappa $ . In addition, $m\leq k$ and we have:

1. (1) $|v_{i}|\ll _{k,\kappa } R_{0}\overline {l}_{1}^{1-i-\kappa } $ for all $1\leq i\leq k$ ;

2. (2) fix $\eta \approx 0$ . Assume that for certain $1\leq j\leq m-1$ , sufficiently large $\overline {l}_{j}$ , the intervals $[0,\overline {l}_{j}]$ and $[l_{j+1},\overline {l}_{j+1}]$ do not have an effective gap:

   (4.3) $$ \begin{align} l_{j+1}-\overline{l}_{j}\leq \min\{\overline{l}_{j},\overline{l}_{j+1}-l_{j+1}\}^{1+\eta}. \end{align} $$
   Then there exists $1\approx \xi (\eta ,k)\in (0,1)$ with $\xi (\eta ,k)\rightarrow 1$ as $\eta \rightarrow 0$ such that
   $$ \begin{align*}|v_{i}|\ll_{k,\kappa} \overline{l}_{j}^{\xi(\eta,k)(1-i-\kappa)}\end{align*} $$
   for all $1\leq i\leq k$ .

Proof. The number m of intervals in equation (4.2) can be bounded by k via an elementary argument of polynomials.

(1) Let for $x\in [0,1]$ . Then we have

$$ \begin{align*}\left( \begin{array}{@{}c@{}} v_{1}\overline{l}_{1}^{\kappa} \\ v_{2}\overline{l}_{1}^{1+\kappa} \\ \vdots \\ v_{k}\overline{l}_{1}^{k-1+\kappa} \\ \end{array} \right)=\left[ \begin{array}{@{}cccc@{}} (1/k)^{\kappa}& (1/k)^{1+\kappa} & \cdots & (1/k)^{k-1+\kappa} \\ (2/k)^{\kappa}& (2/k)^{1+\kappa} & \cdots & (2/k)^{k-1+\kappa} \\ \vdots& \vdots & \ddots & \vdots \\ 1& 1 & \cdots & 1 \\ \end{array} \right]^{-1}\left( \begin{array}{@{}c@{}} F(1/k) \\ F(2/k) \\ \vdots \\ F(1) \\ \end{array} \right)\!.\end{align*} $$

By equation (4.2), we know that $|F (1/k)|,|F (2/k)|,\ldots ,|F(1)|\ll R_{0}$ . Thus, we obtain $|v_{i}|\ll _{k,\kappa } R_{0} \overline {l}_{1}^{1-i-\kappa } $ for all $1\leq i\leq k$ .

(2) This follows by induction. Assume that the statement holds for $j-1$ . For j, the only difficult situation is when $\overline {l}_{j}\leq l_{j+1}-\overline {l}_{j}$ and $\overline {l}_{j+1}-l_{j+1}\leq l_{j+1}-\overline {l}_{j}$ . If this is the case, then

$$ \begin{align*} \overline{l}_{j+1}= (\overline{l}_{j+1}-l_{j+1})+(l_{j+1}-\overline{l}_{j})+\overline{l}_{j}\leq 3\overline{l}_{j}^{1+\eta}. \end{align*} $$

Thus, by induction hypothesis, we get

$$ \begin{align*}|v_{i}|\ll \overline{l}_{j}^{\xi(\eta,j)(1-i-\kappa)}\ll \overline{l}_{j+1}^{{\xi(\eta,j)}/({1+\eta})(1-i-\kappa)} \end{align*} $$

for all $1\leq i\leq k$ .

4.2 Effective estimates of shearing phenomena

Now we begin to study the shearing between two nearby orbits of time changes of unipotent flows. Let $G=\mathrm {SO}(n,1)$ . First, since all maximal compact subgroups of $C_{G}(U)$ are conjugate, we can assume without loss of generality that $C^{\rho }$ is in the compact group generated by $\mathfrak {k}^{\perp }_{C}$ . Thus, via equations (2.3), (2.4), and (2.8), we consider the decomposition

$$ \begin{align*} \mathfrak{g}=\mathfrak{sl}_{2}\oplus V^{\perp\rho}\oplus\operatorname{\mathrm{Lie}}(C^{\rho}),\quad V^{\perp\rho}=\sum_{i} V_{i}^{0\perp\rho}\oplus\sum_{j} V_{j}^{2}, \end{align*} $$

$$ \begin{align*} \mathfrak{k}^{\perp}_{C}=\mathfrak{k}^{\perp\rho}_{C}\oplus\operatorname{\mathrm{Lie}}(C^{\rho}),\end{align*} $$

where $\operatorname {\mathrm {Lie}}(C^{\rho })$ denotes the Lie algebra of $C^{\rho }$ and note that $\operatorname {\mathrm {Lie}}(C^{\rho })$ consists of weight $0$ spaces. Since $C^{\rho }$ is compact, there is a G-right invariant metric $d_{C^{\rho }\backslash G}(\cdot ,\cdot )$ on $C^{\rho }\backslash G$ . Let $P:G\rightarrow C^{\rho }\backslash G$ be the natural projection

Then, for $g_{x},g_{y}\in G$ , we have

$$ \begin{align*}d_{C^{\rho}\backslash G}(\overline{g_{x}},\overline{g_{y}})=d_{C^{\rho}\backslash G}( C^{\rho}g_{x} ,C^{\rho}g_{y})=d_{C^{\rho}\backslash G}(C^{\rho}g_{x}g_{y}^{-1},C^{\rho})=d_{C^{\rho}\backslash G}(\overline{g_{x}g_{y}^{-1}},\overline{e}).\end{align*} $$

Moreover, $dP$ induces an isometry between $\mathfrak {sl}_{2}+ V^{\perp \rho }$ and $T_{\overline {e}}(C^{\rho }\backslash G)$ . See for example [Reference Gallier and QuaintanceGQ19] for more details.

Assume $\overline {g}\in B_{C^{\rho }\backslash G}(e,\epsilon )$ for sufficiently small $0<\epsilon $ . Since $C^{\rho }$ in fact commutes with $\mathrm {SO}_{0}(2,1)$ , we can identify

(4.4) $$ \begin{align} \overline{g}=C^{\rho}h\exp v \end{align} $$

for some $h\in B_{\mathrm {SO}_{0}(2,1)}(e,\epsilon )$ and $v\in B_{V^{\perp \rho }}(0,\epsilon )$ . In addition, for $h=[ \begin {smallmatrix} a& b \\ c& d\\ \end {smallmatrix} ]\in B_{\mathrm {SO}_{0}(2,1)}(e,\epsilon )$ , we must have $|b|,|c|<\epsilon $ , $1-\epsilon <|a|,|d|<1+\epsilon $ .

Next, let $t(s)\in \mathbb {R}^{+}$ be a function of $s\in \mathbb {R}^{+}$ . Then we want to study the difference $ u^{t}\overline {g}u^{-s}$ of two nearby orbits of time changes of unipotent flows. By equation (2.5), we have

(4.5) $$ \begin{align} u^{t}\overline{g}u^{-s} &=C^{\rho}u^{t}h\exp v u^{-s}=C^{\rho}(u^{t}hu^{-s})(u^{s}\exp(v)u^{-s}) \nonumber\\ & = C^{\rho}(u^{t}hu^{-s})\exp(\operatorname{\mathrm{Ad}} u^{s}.v) = C^{\rho}(u^{t}hu^{-s})\exp\bigg(\sum_{n=0}^{\varsigma}\sum_{i=0}^{n}b_{i}\binom{n}{i}s^{n-i}v_{n}\bigg). \end{align} $$

Then one may conclude that $u^{t}\overline {g}u^{-s}< \epsilon $ if and only if

(4.6) $$ \begin{align} u^{t}hu^{-s}\ll\epsilon,\quad \operatorname{\mathrm{Ad}} u^{s}.v=\sum_{n=0}^{\varsigma}\sum_{i=0}^{n}b_{i}\binom{n}{i}s^{n-i}v_{n}\ll\epsilon, \end{align} $$

where $\overline {g}\ll \epsilon $ for $g\in G$ means $d_{C^{\rho }\backslash G}(\overline {g},e)\ll \epsilon $ . Therefore, later on, we shall split the elements closing to the identity into two parts, say the $\mathrm {SO}(2,1)$ -part and the $V^{\perp \rho }$ -part.

As shown in equation (4.6), we consider the elements of the form $u^{t}hu^{-s}\in B_{\mathrm {SO}(2,1)}(e,\epsilon )$ . One may calculate

(4.7) $$ \begin{align} u^{t}hu^{-s}= &\left[ \begin{array}{@{}ccc@{}} 1& \\ t& 1\\ \end{array} \!\right]\left[ \begin{array}{@{}ccc@{}} a& b \\ c& d\\ \end{array} \!\right]\left[ \begin{array}{@{}ccc@{}} 1& \\ -s& 1\\ \end{array} \!\right]\;\nonumber\\ =& \left[ \begin{array}{@{}ccc@{}} a-bs& b \\ c+(a-d)s-bs^{2}+(t-s)(a-bs)& d+bt\\ \end{array} \!\right]\!. \; \end{align} $$

If we further impose the Hölder inequality $|s-t|\ll _{\kappa }\max \{R_{0},s^{1-\kappa }\}$ for some $R_{0}>\epsilon $ (see §2.3 or equation (4.33)), then we have the crude estimate

$$ \begin{align*} &|-bs^{2}+(a-d)s+c+(-bs+a)(t-s)|<\epsilon \;\nonumber\\ \Rightarrow\ & |-bs^{2}+(a-d)s|-|c|-|(-bs+a)(t-s)|<\epsilon\;\nonumber\\ \Rightarrow\ & |-bs^{2}+(a-d)s|<2\epsilon+2|t-s|\;\nonumber\\ \Rightarrow\ & |-bs^{2}+(a-d)s|\ll_{\kappa}\max\{R_{0},s^{1-\kappa}\}. \;\nonumber \end{align*} $$

By Lemma 4.3, we immediately obtain the following lemma.

Lemma 4.4. (Estimates for $\mathrm {SO}_{0}(2,1)$ -coefficients)

Given $\kappa \approx 0$ , $R_{0}>0$ , $\epsilon \approx 0$ , a matrix $h=[ \begin {smallmatrix} a& b \\ c& d\\ \end {smallmatrix} ]\in B_{\mathrm {SO}(2,1)}(e,\epsilon )$ , then the solutions $s\in [0,\infty )$ of the following inequality

(4.8) $$ \begin{align} |-bs^{2}+(a-d)s|\ll_{\kappa} \max\{R_{0},s^{1-\kappa}\} \end{align} $$

consist of at most two intervals, say $[0,\overline {l}_{1}(h)]\cup [l_{2}(h),\overline {l}_{2}(h)]$ , where $\overline {l}_{1}$ has the lower bound $l(\epsilon ,R_{0},\kappa )$ such that $l(\epsilon ,R_{0},\kappa )\nearrow \infty $ as $\epsilon \searrow 0$ for fixed $R_{0},\kappa $ . Moreover, we have:

1. (1) $|b|\ll _{\kappa } \overline {l}_{1}^{-1-\kappa }$ and $|a-d|\ll _{\kappa } \overline {l}_{1}^{-\kappa }$ ;

2. (2) if we further assume that the intervals $[0,\overline {l}_{1}]$ and $[l_{2},\overline {l}_{2}]$ do not have an effective gap as in equation (4.3), that is, $l_{2}-\overline {l}_{1}\leq \min \{\overline {l}_{1},\overline {l}_{2}-l_{2}\}^{1+\eta }$ for some $\eta \approx 0$ , then

   $$ \begin{align*}|b|\ll_{\kappa} \overline{l}_{2}^{\xi(\eta)(-1-\kappa)},\quad |a-d|\ll_{\kappa} \overline{l}_{2}^{\xi(\eta)(-\kappa)}.\end{align*} $$

Next, we study the situation when $\operatorname {\mathrm {Ad}} u^{s}.v\ll \epsilon $ . Again by Lemma 4.3, we have the following lemma.

Lemma 4.5. (Estimates for $V^{\perp \rho }$ -coefficients)

Fix $v=b_{0}v_{0}+\cdots +b_{\varsigma }v_{\varsigma }\in B_{V_{\varsigma }}(0,\epsilon )$ . Assume that

$$ \begin{align*} \operatorname{\mathrm{Ad}} u^{s}.v\ll \epsilon \quad\text{if and only if } s\in[0,\overline{l}_{1}(v)] \cup\cdots\cup[l_{m}(v),\overline{l}_{m}(v)], \end{align*} $$

where $\overline {l}_{1}$ has the lower bound $l(\epsilon ,R_{0},\kappa )$ such that $l(\epsilon ,R_{0},\kappa )\nearrow \infty $ as $\epsilon \searrow 0$ for fixed $R_{0},\kappa $ . Then $m=m(v)$ is bounded by a constant depending on $\varsigma $ . Moreover, for $1\leq j\leq \varsigma -1$ , the intervals $[0,\overline {l}_{j}]$ and $[l_{j+1},\overline {l}_{j+1}]$ do not have an effective gap as in equation (4.3), that is, $l_{j+1}-\overline {l}_{j}\leq \min \{\overline {l}_{j},\overline {l}_{j+1}-l_{j+1}\}^{1+\eta }$ , then we have

$$ \begin{align*} |b_{i}|\ll_{\varsigma,\kappa} \overline{l}_{j}^{\xi(\eta,\varsigma)(-\varsigma+i)}. \end{align*} $$

Next, we shall combine the results of Lemmas 4.4 and 4.5. The basic idea is to consider the intersection of the collections of intervals obtained from the above lemmas. For simplicity, we assume that the ‘ $V^{\perp \rho }$ -part’ consists of a single $\mathfrak {sl}_{2}$ -irreducible representation. For the general case, we can repeat the argument for each $\mathfrak {sl}_{2}$ -irreducible representation (cf. §2.2). First, for $\overline {g}=C^{\rho }h\exp (v)\in C^{\rho }\backslash G$ , we write as in Lemmas 4.4 and 4.5

$$ \begin{align*} u^{t}hu^{-s}\ll\epsilon &\text{ if and only if } s\in [0,\overline{l}_{1}(h)]\cup[l_{2}(h),\overline{l}_{2}(h)],\;\nonumber\\ Ad u^{s}.v\ll \epsilon &\text{ if and only if } s\in[0,\overline{l}_{1}(v)]\cup\cdots\cup[l_{m(v)}(v),\overline{l}_{m(v)}(v)].\; \nonumber \end{align*} $$

Write $l_{1}(h)=l_{1}(v)=0$ and we shall consider the family of intervals

(4.9)

where $\overline {l}_{k}(g)<l_{k+1}(g)$ for all k. Thus, in particular, $l_{1}(g)=0$ and $[0,\overline {l}_{1}(g)]=[0,\overline {l}_{1}(h)]\cap [0,\overline {l}_{1}(v)]$ .

Now assume that there exists k such that $[0,\overline {l}_{k}(g)]$ and $[l_{k+1}(g),\overline {l}_{k+1}(g)]$ do not have an effective gap as in equation (4.3), that is,

$$ \begin{align*} l_{k+1}(g)-\overline{l}_{k}(g)\leq \min\{\overline{l}_{k}(g),\overline{l}_{k+1}(g)-l_{k+1}(g)\}^{1+\eta}. \end{align*} $$

Then by Remark 4.2, the corresponding ‘ $\mathrm {SO}(2,1)$ -part’ and ‘ $V^{\perp \rho }$ -part’ should not have effective gaps either. More precisely, for the $\mathrm {SO}(2,1)$ -part, we define

Thus, we know

$$ \begin{align*}[0,\overline{l}_{k}(g)]\subset[0,\overline{l}_{i_{\geq k}}(h)],\quad [l_{k+1}(g),\overline{l}_{k+1}(g)]\subset[l_{i_{\leq k+1}}(h),\overline{l}_{i_{\leq k+1}}(h)] \end{align*} $$

and hence $[0,\overline {l}_{i_{\geq k}}(h)]$ and $[l_{i_{\leq k+1}}(h),\overline {l}_{i_{\leq k+1}}(h)] $ do not have an effective gap as in equation (4.3). Similarly, for the $V^{\perp \rho }$ -part, we define

Then we know

$$ \begin{align*}[0,\overline{l}_{k}(g)]\subset[0,\overline{l}_{j_{\geq k}}(v)],\quad [l_{k+1}(g),\overline{l}_{k+1}(g)]\subset[l_{j_{\leq k+1}}(v),\overline{l}_{j_{\leq k+1}}(v)] \end{align*} $$

and hence $[0,\overline {l}_{j_{\geq k}}(v)]$ and $[l_{j_{\leq k+1}}(v),\overline {l}_{j_{\leq k+1}}(v)] $ do not have an effective gap as in equation (4.3). Further, one observes

$$ \begin{align*} [0,\overline{l}_{k}(g)]&=[0,\overline{l}_{i_{\geq k}}(h)]\cap [0,\overline{l}_{j_{\geq k}}(v)],\;\nonumber\\ [l_{k+1}(g),\overline{l}_{k+1}(g)]&= [l_{i_{\leq k+1}}(h),\overline{l}_{i_{\leq k+1}}(h)]\cap [l_{j_{\leq k+1}}(v),\overline{l}_{j_{\leq k+1}}(v)].\; \nonumber \end{align*} $$

Now recall by the definition in equation (4.9) that the number of intervals in $\{[l_{k}(g),\overline {l}_{k}(g)]\}_{k}$ is bounded by a constant $c(\varsigma )>0$ because the numbers of intervals $\{[l_{i}(h),\overline {l}_{i}(h)]\}_{i}$ , $\{[l_{j}(v),\overline {l}_{j}(v)]\}_{j}$ are. Since $\varsigma \leq 2$ when $\mathfrak {g}=\mathfrak {so}(n,1)$ , we see that $c(\varsigma )$ is uniformly bounded for all $\varsigma $ . Thus, we conclude that the number of intervals in $\{[l_{k}(g),\overline {l}_{k}(g)]\}_{k}$ is uniformly bounded for all $g\in G$ . Then, combining Lemmas 4.5 and 4.4, we obtain the following lemma.

Lemma 4.6. (Estimates for $C^{\rho }\backslash G$ -coefficients)

Let $\kappa \approx 0$ , $R_{0}>0$ , $\epsilon \approx 0$ , $\overline {g}=C^{\rho }h\exp v\in B_{C^{\rho }\backslash G}(e,\epsilon )$ be as above, where

$$ \begin{align*}h=\left[ \begin{array}{@{}ccc@{}} a& b \\ c& d\\ \end{array} \!\!\right]\in \mathrm{SO}_{0}(2,1),\quad v=b_{0}v_{0}+\cdots+b_{\varsigma}v_{\varsigma}\in V_{\varsigma}.\end{align*} $$

Next, let $t(s)\in \mathbb {R}^{+}$ be a function of $s\in \mathbb {R}^{+}$ which satisfies the effectiveness

$$ \begin{align*}|s-t(s)|\ll_{\kappa}\max\{R_{0},s^{1-\kappa}\}.\end{align*} $$

Then there exist intervals $\{[l_{k}(g),\overline {l}_{k}(g)]\}_{k}$ such that

(4.10) $$ \begin{align} u^{t}\overline{g}u^{-s}<\epsilon, \quad\text{implies}\ s\in \bigcup_{k}[l_{k}(g),\overline{l}_{k}(g)], \end{align} $$

where $\overline {l}_{1}$ has the lower bound $l(\epsilon ,R_{0},\kappa )$ such that $l(\epsilon ,R_{0},\kappa )\nearrow \infty $ as $\epsilon \searrow 0$ for fixed $R_{0},\kappa $ . In addition, $k\leq c$ for some constant $c=c(\mathfrak {g})>0$ , and:

1. (1) $|b|\ll _{\kappa } \overline {l}_{1}(g)^{-1-\kappa }$ , $|a-d|\ll _{\kappa } \overline {l}_{1}(g)^{-\kappa }$ , $|b_{i}|\ll _{\varsigma ,\kappa } \overline {l}_{1}(g)^{-\varsigma +i}$ for all $0\leq i\leq \varsigma $ ;

2. (2) If we further assume that the intervals $[0,\overline {l}_{k}(g)]$ and $[l_{k+1}(g),\overline {l}_{k+1}(g)]$ do not have an effective gap as in equation (4.3), then there exists $1\approx \xi =\xi (\eta )\in (0,1)$ with $\xi \rightarrow 1$ as $\eta \rightarrow 0$ such that

   $$ \begin{align*}|b|\ll_{\kappa} \overline{l}_{k}(g)^{-\xi(1+\kappa)},\quad |a-d|\ll_{\kappa} \overline{l}_{k}(g)^{-\xi\kappa},\quad |b_{i}|\ll_{\varsigma,\kappa} \overline{l}_{k}(g)^{-\xi (\varsigma-i)}\end{align*} $$
   for all $1\leq i\leq \varsigma $ .

In practical use, we consider two strictly increasing functions $t(r),s(r)\in \mathbb {R}^{+}$ of $r\in \mathbb {R}^{+}$ satisfying the effective estimates

(4.11) $$ \begin{align} |r-t(r)|\ll_{\kappa}\max\{R_{0},r^{1-\kappa}\},\quad |r-s(r)|\ll_{\kappa}\max\{R_{0},r^{1-\kappa}\}. \end{align} $$

It follows that t is also an increasing function of s and satisfies

$$ \begin{align*}|t(r)-s(r)|\leq|t(r)-r|+|r-s(r)|\ll_{\kappa}\max\{R_{0},r^{1-\kappa}\}\ll_{\kappa}\max\{R_{0},s(r)^{1-\kappa}\}.\end{align*} $$

Then by Lemma 4.6 and the monotonic nature, we deduce the following corollary.

Corollary 4.7. (Change of variables)

Let $\kappa \approx 0$ , $R_{0}>0$ , $\epsilon \approx 0$ , $\overline {g}=C^{\rho }h\exp v\in B_{C^{\rho }\backslash G}(e,\epsilon )$ be as above, where

$$ \begin{align*}h=\left[ \begin{array}{@{}ccc@{}} a& b \\ c& d\\ \end{array} \!\!\right]\in \mathrm{SO}_{0}(2,1),\quad v=b_{0}v_{0}+\cdots+b_{\varsigma}v_{\varsigma}\in V_{\varsigma}.\end{align*} $$

Assume that we have equation (4.11). Then there exist intervals $\{[l_{k}(g),\overline {l}_{k}(g)]\}_{k}$ such that

(4.12) $$ \begin{align} u^{t(r)}\overline{g}u^{-s(r)}<\epsilon \quad\text{implies }\, r\in \bigcup_{k}[L_{k}(g),\overline{L}_{k}(g)], \end{align} $$

where $\overline {L}_{1}$ has the lower bound $L(\epsilon ,R_{0},\kappa )$ such that $L(\epsilon ,R_{0},\kappa )\nearrow \infty $ as $\epsilon \searrow 0$ for fixed $R_{0},\kappa $ . Then we have $k\leq c$ for some constant $c=c(\mathfrak {g})>0$ , and:

1. (1) $|b|\ll _{\kappa } \overline {L}_{1}(g)^{-1-\kappa }$ , $|a-d|\ll _{\kappa } \overline {L}_{1}(g)^{-\kappa }$ , $|b_{i}|\ll _{\varsigma ,\kappa } \overline {L}_{1}(g)^{-\varsigma +i}$ for all $0\leq i\leq \varsigma $ ;

2. (2) if we further assume that the intervals $[0,\overline {L}_{k}(g)]$ and $[L_{k+1}(g),\overline {L}_{k+1}(g)]$ do not have an effective gap, as in equation (4.3), then there exists $1\approx \xi =\xi (\eta )\in (0,1)$ with $\xi \rightarrow 1$ as $\eta \rightarrow 0$ such that

   $$ \begin{align*}|b|\ll_{\kappa} \overline{L}_{k}(g)^{-\xi(1+\kappa)},\quad |a-d|\ll_{\kappa} \overline{L}_{k}(g)^{-\xi\kappa},\quad |b_{i}|\ll_{\varsigma,\kappa} \overline{L}_{k}(g)^{-\xi (\varsigma-i)}\end{align*} $$
   for all $1\leq i\leq \varsigma $ .

4.3 $\epsilon $ -blocks and effective gaps

Let $x\in \overline {X}$ , $y\in B_{\overline {X}}(x,\epsilon )$ . We say that $(\overline {g_{x}},\overline {g_{y}})\in C^{\rho }\backslash G\times C^{\rho }\backslash G$ covers $(x,y)$ if $ d_{C^{\rho }\backslash G}(\overline {g_{x}},\overline {g_{y}})<\epsilon $ and $\overline {P}(\overline {g_{x}})=x$ , $\overline {P}(\overline {g_{y}})=y$ , where $\overline {P}:C^{\rho }\backslash G\rightarrow C^{\rho }\backslash G/\Gamma $ is the projection. Since $\operatorname {\mathrm {Lie}}(C^{\rho }\backslash G)\cong \mathfrak {sl}_{2}+ V^{\perp \rho }$ , given a representative $g_{x}$ of $\overline {g_{x}}$ , we may choose $g_{y}\in G$ such that $P(g_{y})=\overline {g_{y}}$ and

$$ \begin{align*} \log (g_{y}g_{x}^{-1})\in \mathfrak{sl}_{2}+ V^{\perp\rho}. \end{align*} $$

We shall always make such a choice if no further explanation.

Definition 4.2. ( $\epsilon $ -block)

Suppose that $x\in \overline {X}$ , $y\in B_{\overline {X}}(x,\epsilon )$ , $(\overline {g_{x}},\overline {g_{y}})$ covers $(x,y)$ , and $R\in (0,\infty ]$ satisfies

$$ \begin{align*} d_{C^{\rho}\backslash G}(u^{s(R)}\overline{g_{x}},u^{t(R)}\overline{g_{y}})<\epsilon. \end{align*} $$

Then we define the $\epsilon $ -block of $\overline {g_{x}},\overline {g_{y}}$ of length r by

Similarly, we define the $\epsilon $ -block of $x,y$ of length r by

In either case, we call $[0,R]$ the corresponding time interval and define the length $|\operatorname {\mathrm {BL}}|$ of $\operatorname {\mathrm {BL}}$ by

We also write

$$ \begin{align*}\operatorname{\mathrm{BL}}(x,y)=\{(x,y),(u^{s(R)}x ,u^{t(R)}y)\}=\{(x,y),(\overline{x},\overline{y})\}\end{align*} $$

emphasizing that $(x,y)$ is the first and $(\overline {x},\overline {y})$ is the last pair of the block $\operatorname {\mathrm {BL}}(x,y)$ .

For a pair of $\epsilon $ -blocks, a shifting problem may occur.

Definition 4.3. (Shifting)

Let $\overline {\operatorname {\mathrm {BL}}}^{\prime }=\{(x^{\prime },y^{\prime }),(\overline {x}^{\prime },\overline {y}^{\prime })\}$ , $\overline {\operatorname {\mathrm {BL}}}^{\prime \prime }=\{(x^{\prime \prime },y^{\prime \prime }),(\overline {x}^{\prime \prime },\overline {y}^{\prime \prime })\}$ be two $\epsilon $ -blocks. Then $x^{\prime \prime }= u^{s}g_{x^{\prime }}$ , $y^{\prime \prime }= u^{t}y^{\prime }$ for some $s,t>0$ . Further, there is a unique $\gamma \in \Gamma $ such that

(4.13) $$ \begin{align} d_{C^{\rho}\backslash G}(\overline{g_{x^{\prime\prime}}},\overline{g_{y^{\prime\prime}}}\gamma)<\epsilon, \end{align} $$

where , . We define:

• • (Shifting) $(x^{\prime },y^{\prime })\overset {\Gamma }{\sim }(x^{\prime \prime },y^{\prime \prime })$ if $\gamma \neq e$ in equation (4.13);

• • (Non-shifting) $(x^{\prime },y^{\prime })\overset {e}{\sim }(x^{\prime \prime },y^{\prime \prime })$ if $\gamma =e$ in equation (4.13).

The key observation here is that whenever the difference of $\overline {g_{x}},\overline {g_{y}}$ can be estimated by the length in an appropriate way, a shifting must lead to an effective gap between two $\epsilon $ -blocks. This follows from the natural renormalization of unipotent flows via diagonal flows.

Proposition 4.8. (Shiftings imply effective gaps)

There are quantities $\eta _{0}\approx 0$ , $\sigma _{0}\approx 0$ , $\epsilon _{0}\approx 0$ , $r_{0}>0$ determined orderly such that, for any

• • $\eta \in (0,\eta _{0})$ ,

• • $\sigma \in (0,\sigma _{0}(\eta ))$ ,

• • $\epsilon \in (0,\epsilon _{0}(\sigma ))$ ,

there exists a compact set $K\subset \overline {X}$ with $\overline {\mu }(K)>1-\sigma $ such that the following holds (see Figure 1).

Figure 1

The solid straight lines are the unipotent orbits in the $\overline {\operatorname {\mathrm {BL}}}^{\prime }$ and $\overline {\operatorname {\mathrm {BL}}}^{\prime \prime }$ , and the dashed lines are the rest of the unipotent orbits. The bent curves indicate the length defined by the letters.

Assume that there are two $\epsilon $ -blocks $\overline {\operatorname {\mathrm {BL}}}^{\prime }=\{(x^{\prime },y^{\prime }),(\overline {x}^{\prime },\overline {y}^{\prime })\}$ , $\overline {\operatorname {\mathrm {BL}}}^{\prime \prime }=\{(x^{\prime \prime },y^{\prime \prime }), (\overline {x}^{\prime \prime },\overline {y}^{\prime \prime })\}$ such that the y-endpoints lie in K (that is, $y^{\prime },\overline {y}^{\prime },y^{\prime \prime },\overline {y}^{\prime \prime } \in K$ ) and satisfy

(4.14) $$ \begin{align} g_{y^{\prime}}= h^{\prime}\exp(v^{\prime})g_{x^{\prime}},\quad g_{y^{\prime\prime}}= h^{\prime\prime}\exp(v^{\prime\prime})g_{x^{\prime\prime}}, \end{align} $$

where $h^{\prime }, h^{\prime \prime } \in \mathrm {SO}_{0}(2,1)$ , $v^{\prime }, v^{\prime \prime } \in V_{\varsigma }$ can be estimated by

(4.15) $$ \begin{align} h^{\prime} ,h^{\prime\prime}=\left[ \begin{array}{@{}ccc@{}} 1+O(r^{-2\eta}) & O(r^{-1-2\eta}) \\ O(\epsilon) & 1+O(r^{-2\eta})\\ \end{array} \!\!\right]\!,\quad v^{\prime}, v^{\prime\prime} =O(r^{-\xi\varsigma})v_{0}+\cdots+O(\epsilon)v_{\varsigma} \end{align} $$

for some $r> r_{0}(\sigma ,\epsilon _{0})$ , where $\xi =\xi (\eta )\approx 1$ is given by Corollary 4.7. Assume further that $x^{\prime \prime }=u^{s}\overline {x}^{\prime }$ , $y^{\prime \prime }=u^{t}\overline {y}^{\prime }$ , and $t\asymp s$ . If $\overline {\operatorname {\mathrm {BL}}}^{\prime }\overset {\Gamma }{\sim }\overline {\operatorname {\mathrm {BL}}}^{\prime \prime }$ , then

(4.16) $$ \begin{align} s,t> r^{1+\eta}. \end{align} $$

Proof. We only consider $\varsigma =2$ . Denote

(4.17) $$ \begin{align} g_{\overline{y}^{\prime}}= \overline{h}^{\prime}\exp(\overline{v}^{\prime})g_{\overline{x}^{\prime}} \end{align} $$

for $\overline {h}^{\prime } \in \mathrm {SO}_{0}(2,1)$ , $\overline {v}^{\prime } \in V_{2}$ . By Definition 4.2, we know that $g_{\overline {y}^{\prime }}, g_{\overline {x}^{\prime }}$ are obtained by the unipotent action on $g_{y^{\prime }}, g_{x^{\prime }}$ , and the difference of $g_{\overline {y}^{\prime }}, g_{\overline {x}^{\prime }}$ is controlled by $\epsilon $ . Combining equation (4.15), we get that

(4.18) $$ \begin{align} \overline{h}^{\prime} =\left[ \begin{array}{@{}ccc@{}} 1+O(\epsilon) & O(r^{-1-2\eta}) \\ O(\epsilon) & 1+O(\epsilon)\\ \end{array} \!\!\right]\!,\quad \overline{v}^{\prime} =O(r^{-2\xi })v_{0}+O(\epsilon)v_{1}+ O(\epsilon)v_{2}. \end{align} $$

Since $\overline {\operatorname {\mathrm {BL}}}^{\prime }\overset {\Gamma }{\sim }\overline {\operatorname {\mathrm {BL}}}^{\prime \prime }$ and $g_{x^{\prime \prime }}=u^{s}g_{\overline {x}^{\prime }}$ , we get that

(4.19) $$ \begin{align} g_{y^{\prime\prime}}= cu^{t}g_{\overline{y}^{\prime}}\gamma\quad \text{for some }e\neq \gamma\in\Gamma,\ c\in C^{\rho}. \end{align} $$

Then by equations (4.14), (4.17), and (4.19), we have

(4.20) $$ \begin{align} g_{\overline{y}^{\prime}}&= \overline{h}^{\prime}\exp (\overline{v}^{\prime}) u^{-s} g_{x^{\prime\prime}},\nonumber\\ g_{\overline{y}^{\prime}} \gamma &= c^{-1} u^{-t} h^{\prime\prime}\exp (v^{\prime\prime}) g_{x^{\prime\prime}}. \end{align} $$

Assume that one of $s,t$ is not greater than $r^{1+\eta }$ . Then since $s\asymp t$ , we know

(4.21) $$ \begin{align} 0<s,t\leq O(r^{1+\eta}). \end{align} $$

Next, we determine the quantities for the proposition.

• • (Choice of $\eta $ , $\delta $ (also $\eta _{0}$ )) Choose a small $\eta \approx 0$ that satisfies

  (4.22) $$ \begin{align} 1+2\delta<1+2\eta<2\xi(2\eta), \end{align} $$
  where $\xi (2\eta )$ was defined in Corollary 4.7, and . Here, $\eta _{0}\approx 0$ can be defined to be the maximal $\eta $ so that equation (4.22) holds.

• • (Choice of $\sigma $ ) Then $\sigma =\sigma (\eta )>0$ can be chosen as

  (4.23) $$ \begin{align} \sigma<\frac{3\eta}{4+6\eta}. \end{align} $$

• • (Choice of $\epsilon _{0}$ , $K_{1}$ ; injectivity radius) Since $\Gamma $ is discrete, there is a compact subset $K_{1}\subset \overline {X}$ , $\overline {\mu }(K_{1})>1- \tfrac 14\sigma $ and $ \epsilon _{0}>0$ such that for any $\overline {g_{y}}\in \overline {P}^{-1}(K_{1})$ satisfying

  (4.24) $$ \begin{align} d_{C^{\rho}\backslash G}( \overline{g_{y}}, \overline{g_{y}}\gamma)<O(\epsilon_{0}) \end{align} $$
  for some $\gamma \in \Gamma $ , then $\gamma =e$ . Here the constants hidden in $O(\epsilon _{0})$ will be determined after the estimate of equation (4.28) (see also equation (4.29)).

• • (Choice of $K_{2}$ , K, $T_{0}$ , $r_{0}$ ; ergodicity of $a^{T}$ ) Since the diagonal action $a^{T}$ is ergodic on $(\overline {X},\overline {\mu })$ , there is a compact subset $K_{2}\subset \overline {X}$ , $\overline {\mu }(K_{2})>1-\tfrac 14\sigma $ and $T_{0}=T_{0}(K_{2})>0$ such that the relative length measure $K_{2}$ on $[y,a^{T}y]$ (and $[a^{-T}y,y]$ ) is greater than $1-\sigma $ for any $y\in K_{2}$ , $|T|\geq T_{0}$ . Assume that

  (4.25)

  
  Note that $\overline {\mu }(K)>1-\sigma $ . The quantity $r_{0}$ will be even larger and determined by $\epsilon _{0}$ if necessary (see equation (4.29)).

Now we are in the position to apply the renomalization via the diagonal action $a^{w}$ . Since $r>r_{0}=e^{(1+2\delta )^{-1}T_{0}}$ , let and we know $\omega _{0}>T_{0}$ . Since $\overline {y}^{\prime }\in K\subset K_{2}$ , it follows from the choice of $K_{2}$ and $T_{0}$ that the relative length measure of K on $[\overline {y}^{\prime },a^{\omega _{0}}\overline {y}^{\prime }]$ is greater than $1-\sigma $ . This implies that there is $\omega $ satisfying

$$ \begin{align*} (1-\sigma)\omega_{0}<\omega\leq\omega_{0} \end{align*} $$

such that $a^{\omega }\overline {y}^{\prime }\in K$ and therefore

(4.26) $$ \begin{align} a^{\omega}\overline{g_{\overline{y}^{\prime}}}\in \overline{P}^{-1}(K). \end{align} $$

By equation (4.20), we have

(4.27) $$ \begin{align} a^{\omega} g_{\overline{y}^{\prime}} &= (a^{\omega} \overline{h}^{\prime} a^{-\omega})\exp (\operatorname{\mathrm{Ad}} a^{\omega}. \overline{v}^{\prime}) (a^{\omega}u^{-s}a^{-\omega}) a^{\omega} g_{x^{\prime\prime}},\nonumber\\ a^{\omega} g_{\overline{y}^{\prime}}\gamma &= c^{-1}(a^{\omega}u^{-t} a^{-\omega}) (a^{\omega} h^{\prime\prime} a^{-\omega})\exp (\operatorname{\mathrm{Ad}} a^{\omega}.v^{\prime\prime}) a^{\omega}g_{x^{\prime\prime}}. \end{align} $$

Then by equations (4.18), (4.15), and (4.21), we estimate

(4.28) $$ \begin{align} a^{w} \overline{h}^{\prime}a^{-w}&= \left[ \begin{array}{@{}ccc@{}} 1+O(\epsilon) & O(r^{2\delta-2\eta}) \\ O(\epsilon) & 1+O(\epsilon)\\ \end{array} \!\!\right]\! ,\nonumber\\[2pt] a^{w} \overline{h}^{\prime}a^{-w}&= \left[ \begin{array}{@{}ccc@{}} 1+O(r^{-2\eta}) & O(r^{2\delta-2\eta}) \\ O(\epsilon) & 1+O(r^{-2\eta})\\ \end{array} \!\!\right]\!,\nonumber\\[2pt] \operatorname{\mathrm{Ad}} a^{\omega}. \overline{v}^{\prime}&= O(r^{ -2\xi+1+2\delta })v_{0}+O(\epsilon)v_{1}+O(\epsilon)v_{2},\nonumber\\[2pt] \operatorname{\mathrm{Ad}} a^{\omega}. v^{\prime\prime}&= O(r^{ -2\xi+1+2\delta })v_{0}+O(\epsilon)v_{1}+O(r^{ - (1-\sigma)(1+2\delta) })v_{2} ,\\[2pt] a^{\omega}u^{-t}a^{-\omega}&= u^{-te^{-\omega}}= u^{O(r^{1+\eta}r^{-(1-\sigma)(1+2\delta)})},\nonumber\\[2pt] a^{\omega}u^{-s}a^{-\omega}&= u^{-se^{-\omega}}= u^{O(r^{1+\eta}r^{-(1-\sigma)(1+2\delta)})}. \nonumber \end{align} $$

Notice that by the choice of $\sigma ,\delta $ (see equations (4.22) and (4.23)), we have

$$ \begin{align*}1+\eta-(1-\sigma)(1+2\delta)=1+\eta-(1-\sigma)\big(1+\tfrac{3}{2}\eta\big)<-\tfrac{1}{4}\eta.\end{align*} $$

Also, by equation (4.22), we have

$$ \begin{align*}2\delta-2\eta<0, -2\xi+1+2\delta<0.\end{align*} $$

Thus, by enlarging $r_{0}$ if necessary, all terms of equation (4.28) can be quantitatively dominated by $O(\epsilon _{0})$ . Then by equation (4.27), we have

(4.29) $$ \begin{align} d_{C^{\rho}\backslash G}(\overline{a^{\omega} g_{\overline{y}^{\prime}}}\gamma,\overline{a^{\omega} g_{\overline{y}^{\prime}}})=d_{C^{\rho}\backslash G}(\overline{a^{\omega} g_{\overline{y}^{\prime}}}\gamma (a^{\omega} g_{x^{\prime\prime}})^{-1},\overline{a^{\omega} g_{\overline{y}^{\prime}}}(a^{\omega} g_{x^{\prime\prime}})^{-1})<O(\epsilon_{0}). \end{align} $$

Thus, by equation (4.24), we get $\gamma =e$ , which contradicts our assumptions.

4.4 Construction of $\epsilon $ -blocks

In light of Proposition 4.8, we try to construct a collection of $\epsilon $ -blocks based on the unipotent flows between two nearby points so that each pair of $\epsilon $ -blocks has an effective gap.

First, given $\eta _{0}\approx 0$ as in Proposition 4.8, we fix a sufficiently small $\kappa \in (0,2\eta _{0})$ , and then choose $\eta =\eta (\kappa )\approx 0$ such that

(4.30) $$ \begin{align} \frac{1+2\eta}{\xi(2\eta)}<1+\kappa<1+2\eta_{0}, \end{align} $$

where $\xi (2\eta )\approx 1$ is given by Corollary 4.7. Then, $\sigma _{0}=\sigma _{0}(\eta )\approx 0$ given in Proposition 4.8 has been determined. Next, assume that there exist:

• • $\sigma \in (0,\sigma _{0})$ ;

• • $R_{0}>1$ ;

• • $\epsilon _{0}=\epsilon _{0}(\sigma )\approx 0$ , $\epsilon =\epsilon (R_{0})\in (0,\epsilon _{0})$ so small that

  (4.31) $$ \begin{align} \overline{L}_{1}(g)\geq L(\epsilon,R_{0},\kappa)>\max\{r_{0}(\sigma,\epsilon_{0}),R_{0}\} \end{align} $$
  whenever $g\in B_{G}(e,\epsilon )$ , where $\overline {L}_{1},L$ are defined by Corollary 4.7,

such that, for $K\subset \overline {X}$ with $\overline {\mu }(K)>1-\sigma $ given by Proposition 4.8, $x,y\in \overline {X}$ , we have $A=A(x,y)\subset \mathbb {R}^{+}$ such that:

1. (1) if $r\in A$ , then

   (4.32) $$ \begin{align} u^{t(r)}y\in K\quad\text{and}\quad d_{\overline{X}}(u^{s(r)}x, u^{t(r)}y)<\epsilon \end{align} $$
   for continuous increasing functions $t,s:[0,\infty )\rightarrow [0,\infty )$ ;

2. (2) we have the Hölder inequalities:

   (4.33) $$ \begin{align} |(t(r^{\prime})-t(r))-(r^{\prime}-r)|\ll& |r^{\prime}-r|^{1-\kappa},\\ |(s(r^{\prime})-s(r))-(r^{\prime}-r)|\ll& |r^{\prime}-r|^{1-\kappa}, \nonumber \end{align} $$
   for all $r,r^{\prime }\in A$ with $r^{\prime }>r$ , $ r^{\prime }-r \geq R_{0}$ .

It is worth noting from equation (4.24) that points in K have injectivity radius at least $\epsilon _{0}$ . For simplicity, we shall assume that $0\in A$ in what follows.

4.4.1 Construction of $\beta _{1}$

For $\unicode{x3bb} \in A$ , denote

. Now we construct a collection $\beta _{1}(A_{\unicode{x3bb} })$ of $\epsilon $ -blocks. Let

,

. We follow the assumptions in equations (4.32) and (4.33). Suppose that $(\overline {g_{x_{1}}}, \overline {g_{y_{1}}})\in C^{\rho }\backslash G\times C^{\rho }\backslash G$ covers $(x_{1},y_{1})$ and

where $\overline {L}_{1}$ is defined by Corollary 4.7. Let $\operatorname {\mathrm {BL}}_{1}$ be the $\epsilon $ -block of $x_{1},y_{1}$ of length $\overline {r}_{1}$ , $\operatorname {\mathrm {BL}}_{1}=\{(x_{1},y_{1}),(\overline {x}_{1},\overline {y}_{1})\}$ . To define $\operatorname {\mathrm {BL}}_{2}$ , we take

and apply the above procedure to

(note that by equation (4.12), $r_{2}>\overline {r}_{1}$ ). This process defines a collection $\beta _{1}(A_{\unicode{x3bb} })=\{\operatorname {\mathrm {BL}}_{1},\ldots , \operatorname {\mathrm {BL}}_{n}\}$ of $\epsilon $ -blocks on the orbit intervals $[x_{1},u^{s(\unicode{x3bb} )}x_{1}]$ , $[y_{1},u^{t(\unicode{x3bb} )}y_{1}]$ (see Figure 2):

$$ \begin{align*}x_{i}= u^{s_{i}}x_{1},\quad \overline{x}_{i}=u^{\overline{s}_{i}}x_{1}, \quad y_{i}=u^{t_{i}} y_{1},\quad \overline{y}_{i}=u^{\overline{t}_{i}}y_{1},\end{align*} $$

$$ \begin{align*}s_{i}= s(r_{i}),\quad \overline{s}_{i}= s(\overline{r}_{i}), \quad t_{i}= t(r_{i}),\quad \overline{t}_{i}= t(\overline{r}_{i}).\end{align*} $$

Note also that by the assumption of A, we have $x_{i},\overline {x}_{i}\in K$ for all i, the corresponding time interval of $BL_{i}$ is $[r_{i},\overline {r}_{i}]$ , and the length $|\operatorname {\mathrm {BL}}_{i}|$ of $\operatorname {\mathrm {BL}}_{i}$ is

Figure 2

A collection of $\epsilon $ -blocks $\{\operatorname {\mathrm {BL}}_{1},\ldots ,\operatorname {\mathrm {BL}}_{n}\}$ . The solid straight lines are the unipotent orbits in the $\epsilon $ -blocks and the dashed lines are the rest of the unipotent orbits. The bent curves indicate the length defined by the letters.

Note that any $\operatorname {\mathrm {BL}}_{i}=\{(x_{i},y_{i}),(\overline {x}_{i},\overline {y}_{i})\}\in \beta _{1}(A_{\unicode{x3bb} })$ has length $|\operatorname {\mathrm {BL}}_{i}|\leq \overline {L}_{1}(g_{y_{i}}g_{x_{i}}^{-1})$ . By Corollary 4.7, we immediately obtain an estimate for the difference of $g_{x_{i}}$ and $g_{y_{i}}$ in terms of the length of $\epsilon $ -blocks.

Corollary 4.10. (Difference of $\beta _{1}(A_{\unicode{x3bb} })$ )

Assume that $\overline {g_{y_{i}}g_{x_{i}}^{-1}}=C^{\rho }h_{i}\exp (v_{i})$ , where

$$ \begin{align*}h_{i}=\left[ \begin{array}{@{}ccc@{}} a& b \\ c& d\\ \end{array} \!\!\right]\in \mathrm{SO}_{0}(2,1),\quad v_{i}=b_{0}v_{0}+\cdots+b_{\varsigma}v_{\varsigma}\in V_{\varsigma}.\end{align*} $$

Then we have

$$ \begin{align*}h_{i}=\left[ \begin{array}{@{}ccc@{}} 1+O(\mathbf{r}_{i}^{-\kappa}) & O(\mathbf{r}_{i}^{-1-\kappa}) \\ O(\epsilon) & 1+O(\mathbf{r}_{i}^{-\kappa})\\ \end{array} \!\!\right]\!,\quad v_{i}=O(\mathbf{r}_{i}^{-\varsigma})v_{0}+\cdots+O(\epsilon)v_{\varsigma}\end{align*} $$

for some $\mathbf {r}_{i}\geq \max \{r_{0},R_{0},|\!\operatorname {\mathrm {BL}}_{i}\!|\}$ .

We then immediately conclude from Proposition 4.8 that for any $\operatorname {\mathrm {BL}}^{\prime },\operatorname {\mathrm {BL}}^{\prime \prime }\in \beta _{1}(A_{\unicode{x3bb} })$ with $\operatorname {\mathrm {BL}}^{\prime }\overset {\Gamma }{\sim }\operatorname {\mathrm {BL}}^{\prime \prime }$ , there is an effective gap between them, that is,

$$ \begin{align*} d(\operatorname{\mathrm{BL}}^{\prime},\operatorname{\mathrm{BL}}^{\prime\prime})\geq[\min\{|\operatorname{\mathrm{BL}}^{\prime}\!|,|\operatorname{\mathrm{BL}}^{\prime\prime}\!|\}]^{1+\kappa/2}. \end{align*} $$

However, when $\operatorname {\mathrm {BL}}^{\prime }\overset {e}{\sim }\operatorname {\mathrm {BL}}^{\prime \prime }$ , they do not necessarily have an effective gap. This enlightens us to connect these $\epsilon $ -blocks and generate a new collection $\beta _{2}(A_{\unicode{x3bb} })$ .

4.4.2 Construction of $\beta _{2}$

Now we construct a new collection $\beta _{2}(A_{\unicode{x3bb} })=\{\overline {\operatorname {\mathrm {BL}}}_{1},\ldots ,\overline {\operatorname {\mathrm {BL}}}_{N}\}$ by the following procedure. The idea is to connect $\epsilon $ -blocks in $\beta _{1}(A_{\unicode{x3bb} })=\{\operatorname {\mathrm {BL}}_{1},\ldots ,\operatorname {\mathrm {BL}}_{n}\}$ so that each pair of new blocks must have an effective gap. Let $\operatorname {\mathrm {BL}}_{1}\in \beta _{1}(A_{\unicode{x3bb} })$ , $g_{y_{1}}=h\exp (v)g_{x_{1}}$ , and

$$ \begin{align*}h=\left[ \begin{array}{@{}ccc@{}} a& b \\ c& d\\ \end{array} \!\!\right]\in \mathrm{SO}(2,1),\quad v=b_{0}v_{0}+\cdots+b_{\varsigma}v_{\varsigma}\in V_{\varsigma}.\end{align*} $$

Then by Corollary 4.7, one can write $u^{t(r)}gu^{-s(r)}\in B_{G}(e,\epsilon )$ for

(4.34) $$ \begin{align} r\in \bigcup_{k}[L_{k}(g),\overline{L}_{k}(g)], \end{align} $$

where $k\leq c$ is uniformly bounded for all $g\in G$ . Then consider the following two cases.

1. (1) There is no $j\in \{2,\ldots ,n\}$ such that $(x_{1},y_{1})\overset {e}{\sim }(x_{j},y_{j})$ .

2. (2) There is $j\in \{2,\ldots ,n\}$ such that $(x_{1},y_{1})\overset {e}{\sim }(x_{j},y_{j})$ .

In case (i), we set $\overline {\operatorname {\mathrm {BL}}}_{1}=\operatorname {\mathrm {BL}}_{1}$ . Then by Corollary 4.10, we have

(4.35) $$ \begin{align} |b|\ll\overline{L}_{1}(g_{y_{1}}g_{x_{1}}^{-1})^{-1-\kappa},\quad |a-d|\leq \overline{L}_{1}(g_{y_{1}}g_{x_{1}}^{-1})^{-\kappa}. \end{align} $$

In case (ii), suppose that $g_{x_{j}}=u^{s_{j}}g_{x_{1}}$ , $g_{y_{j}}=u^{t_{j}}g_{y_{1}}$ . Clearly, by the construction, $\overline {r}_{j}>\overline {L}_{1}(g_{y_{1}}g_{x_{1}}^{-1})$ . However, by equation (4.34), we get

$$ \begin{align*}\overline{r}_{j}\in \bigcup_{k}[L_{k}(g_{y_{1}}g_{x_{1}}^{-1}),\overline{L}_{k}(g_{y_{1}}g_{x_{1}}^{-1})]\end{align*} $$

and $k\leq C$ is uniformly bounded for all $g\in G$ . Assume that $j_{\max }$ is the maximal j among $\overline {r}_{j}\in [L_{2}(g_{y_{1}}g_{x_{1}}^{-1}),\overline {L}_{2}(g_{y_{1}}g_{x_{1}}^{-1})]$ . Whether $[0,\overline {L}_{1}(g_{y_{1}}g_{x_{1}}^{-1})]$ and $[L_{2}(g_{y_{1}}g_{x_{1}}^{-1}),\overline {L}_{2}(g_{y_{1}}g_{x_{1}}^{-1})]$ have an effective gap leads to a dichotomy of choices:

$$ \begin{align*}\overline{\operatorname{\mathrm{BL}}}_{1}=\begin{cases} \text{remains unchanged}&\text{if } L_{2}(g_{y_{1}}g_{x_{1}}^{-1})-\overline{L}_{1}(g_{y_{1}}g_{x_{1}}^{-1})>\overline{L}_{1}(g_{y_{1}}g_{x_{1}}^{-1})^{1+2\eta},\\ \{(x_{1},y_{1}),(\overline{x}_{j_{\max}},\overline{y}_{j_{\max}})\}&\text{otherwise.} \end{cases}\end{align*} $$

If the first case occurs, we will not change $\overline {\operatorname {\mathrm {BL}}}_{1}$ anymore. If the second case occurs, that is, we redefine $\overline {\operatorname {\mathrm {BL}}}_{1}=\{(x_{1},y_{1}),(\overline {x}_{j_{\max }},\overline {y}_{j_{\max }})\}$ , then we repeat the construction for the new $\overline {\operatorname {\mathrm {BL}}}_{1}$ again.

1. (1) Suppose that there is $\overline {r}_{j}>\overline {L}_{2}(g_{y_{1}}g_{x_{1}}^{-1})$ . Then assume $j_{\max }$ to be the maximal j among $\overline {r}_{j}\in [L_{3}(g_{y_{1}}g_{x_{1}}^{-1}),\overline {L}_{3}(g_{y_{1}}g_{x_{1}}^{-1})]$ . Then again, we set

   $$ \begin{align*}\overline{\operatorname{\mathrm{BL}}}_{1}=\begin{cases} \text{remains unchanged}&\!\!\text{if } L_{3}(g_{y}g_{x}^{-1})-\overline{L}_{3}(g_{y_{1}}g_{x_{1}}^{-1})>\overline{L}_{2}(g_{y_{1}}g_{x_{1}}^{-1})^{1+2\eta},\\ \{(x_{1},y_{1}),(\overline{x}_{j_{\max}},\overline{y}_{j_{\max}})\}&\!\!\text{otherwise}, \end{cases} \end{align*} $$
   and so on.

The process will stop since the number of intervals is uniformly bounded for all $g\in G$ . Now $\overline {BL}_{1}\in \beta _{2}(A_{\unicode{x3bb} })$ has been constructed. By the choice of $\overline {\operatorname {\mathrm {BL}}}_{1}$ and Corollary 4.7, we conclude that

(4.36) $$ \begin{align} |b|\ll_{\kappa} |\operatorname{\mathrm{BL}}_{1}|^{-\xi(1+\kappa)},\quad |a-d|\ll_{\kappa} |\operatorname{\mathrm{BL}}_{1}|^{-\xi\kappa},\quad |b_{i}|\ll_{\varsigma,\kappa} |\operatorname{\mathrm{BL}}_{1}|^{-\xi (\varsigma-i)} \end{align} $$

for $\xi =\xi (2\eta )\approx 1$ and for all $1\leq i\leq \varsigma $ .

Next, we repeat the above argument to construct $\overline {\operatorname {\mathrm {BL}}}_{m+1}$ . More precisely, suppose that $ \overline {\operatorname {\mathrm {BL}}}_{m}=\{(x_{j_{m-1}+1},y_{j_{m-1}+1}),(\overline {x}_{j_{m}},\overline {y}_{j_{m}})\}\in \beta _{2}(A_{\unicode{x3bb} })$ has been constructed. To define $\overline {\operatorname {\mathrm {BL}}}_{m+1}$ , we repeat the above argument to $\operatorname {\mathrm {BL}}_{j_{m}+1}\in \beta _{1}(A_{\unicode{x3bb} })$ . Thus, $\beta _{2}(A_{\unicode{x3bb} })$ is completely defined. Further, one may conclude the difference of points of $\epsilon $ -blocks in $\beta _{2}(A_{\unicode{x3bb} })$ .

Lemma 4.11. (Difference of $\beta _{2}(A_{\unicode{x3bb} })$ )

For any $\overline {\operatorname {\mathrm {BL}}}_{i}=\{(x_{i}^{\prime },y_{i}^{\prime }),(\overline {x}_{i}^{\prime },\overline {y}_{i}^{\prime })\}$ in the collection $\beta _{2}(A_{\unicode{x3bb} })=\{\overline {\operatorname {\mathrm {BL}}}_{1},\ldots ,\overline {\operatorname {\mathrm {BL}}}_{N}\}$ of $\epsilon $ -blocks, we have

$$ \begin{align*} \overline{g_{y_{i}^{\prime}}g_{x_{i}^{\prime}}^{-1}}=C^{\rho}h_{i}\exp(v_{i}),\end{align*} $$

where

(4.37) $$ \begin{align} h_{i}=\left[ \begin{array}{@{}ccc@{}} 1+O(\mathbf{r}_{i}^{-2\eta}) & O(\mathbf{r}_{i}^{-1-2\eta}) \\ O(\epsilon) & 1+O(\mathbf{r}_{i}^{-2\eta})\\ \end{array} \!\!\right]\!,\quad v_{i}=O(\mathbf{r}_{i}^{-\xi\varsigma})v_{0}+\cdots+O(\epsilon)v_{\varsigma} \end{align} $$

for some $\mathbf {r}_{i}\geq \max \{r_{0},R_{0},|\overline {\operatorname {\mathrm {BL}}}_{i}|\}$ .

Proof. Equation (4.37) follows immediately from equations (4.35), (4.36), (4.30), and (4.31).

Then, recall that by the construction of $\beta _{2}(A_{\unicode{x3bb} })$ , for any $\overline {\operatorname {\mathrm {BL}}}^{\prime },\overline {\operatorname {\mathrm {BL}}}^{\prime \prime }\in \beta _{2}(A_{\unicode{x3bb} })$ with $\overline {\operatorname {\mathrm {BL}}}^{\prime }\overset {e}{\sim }\overline {\operatorname {\mathrm {BL}}}^{\prime \prime }$ , there is an effective gap between them, that is,

$$ \begin{align*}d(\overline{\operatorname{\mathrm{BL}}}^{\prime},\overline{\operatorname{\mathrm{BL}}}^{\prime\prime})\geq[\max\{r_{0},R_{0},\min\{|\overline{\operatorname{\mathrm{BL}}}^{\prime}|,|\overline{\operatorname{\mathrm{BL}}}^{\prime\prime}|\}\}]^{1+2\eta}.\end{align*} $$

However, when $\overline {\operatorname {\mathrm {BL}}}^{\prime }\overset {\Gamma }{\sim }\overline {\operatorname {\mathrm {BL}}}^{\prime \prime }$ , by Proposition 4.8 and Lemma 4.11, we have

$$ \begin{align*}d(\overline{\operatorname{\mathrm{BL}}}^{\prime},\overline{\operatorname{\mathrm{BL}}}^{\prime\prime})\geq[\max\{r_{0},R_{0},\min\{|\overline{\operatorname{\mathrm{BL}}}^{\prime}|,|\overline{\operatorname{\mathrm{BL}}}^{\prime\prime}|\}\}]^{1+\eta}.\end{align*} $$

Thus, we conclude from Proposition 4.1 the following proposition.

Proposition 4.12. (Effective gaps of $\beta _{2}(A_{\unicode{x3bb} })$ )

Let the notation and assumptions be as above. For any $\overline {\operatorname {\mathrm {BL}}}^{\prime },\overline {\operatorname {\mathrm {BL}}}^{\prime \prime }\in \beta _{2}(A_{\unicode{x3bb} })$ , we have

$$ \begin{align*} d(\overline{\operatorname{\mathrm{BL}}}^{\prime},\overline{\operatorname{\mathrm{BL}}}^{\prime\prime}) \geq[\max\{r_{0},R_{0},\min\{|\overline{\operatorname{\mathrm{BL}}}^{\prime}|,|\overline{\operatorname{\mathrm{BL}}}^{\prime\prime}|\}\}]^{1+\eta}. \end{align*} $$

Thus, for any $\zeta \in [0,1]$ , if

$$ \begin{align*} \frac{1}{\unicode{x3bb}}\operatorname{\mathrm{Leb}}( A_{\unicode{x3bb}})\geq \overline{\theta}_{\eta}(\zeta)=1-\theta(\eta,\zeta)= 1-\prod_{n=0}^{\infty} (1+ C\zeta^{n\eta})^{-1}, \end{align*} $$

then there is an $\epsilon $ -block $\overline {\operatorname {\mathrm {BL}}}\in \beta _{2}(A_{\unicode{x3bb} })$ that has

$$ \begin{align*} |\overline{\operatorname{\mathrm{BL}}}|\geq\zeta\unicode{x3bb}. \end{align*} $$