2 Proof of Theorem 1.2

In this section we prove Theorem 1.2 assuming Theorem  1.4. Let $G,\unicode[STIX]{x1D6E4},F$ be as in Theorem 1.2. We choose and fix a Euclidean norm $\Vert \cdot \Vert$ on the Lie algebra $\mathfrak{g}$ of $G$, which induces a right invariant Riemannian metric $\operatorname{dist}(\cdot ,\cdot )$ on $G$. Moreover, this metric naturally induces a metric on $G/\unicode[STIX]{x1D6E4}$, also denoted by ‘$\operatorname{dist}$’, as follows:

$$\begin{eqnarray}\operatorname{dist}(g\unicode[STIX]{x1D6E4},h\unicode[STIX]{x1D6E4})=\inf _{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}}\operatorname{dist}(g\unicode[STIX]{x1D6FE},h)\quad \text{where }g,h\in G.\end{eqnarray}$$

Let $\mathfrak{r}$ be the maximal amenable ideal of the Lie algebra $\mathfrak{g}$ of $G$, i.e., the largest ideal whose analytic subgroup is amenable. The adjoint action of $G$ on $\mathfrak{s}=\mathfrak{g}/\mathfrak{r}$ defines a homomorphism $\unicode[STIX]{x1D70B}:G\mapsto \operatorname{Aut}(\mathfrak{s})$. Let $S$ be the connected component of $\operatorname{Aut}(\mathfrak{s})$. It follows from the Levi decomposition of $G$ that $\unicode[STIX]{x1D70B}(G)=S$ and $S$ is a center-free semisimple Lie group without compact factors. It is known that $\unicode[STIX]{x1D6E4}\cap \operatorname{Ker}(\unicode[STIX]{x1D70B})$ is a cocompact lattice in $\operatorname{Ker}(\unicode[STIX]{x1D70B})$ and $\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4})$ is a lattice in $S$, see e.g. [Reference Benoist and QuintBQ12, Lemma 6.1]. Therefore, the induced map $\overline{\unicode[STIX]{x1D70B}}:G/\unicode[STIX]{x1D6E4}\mapsto S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4})$ is proper, i.e., the preimage of a compact subset is still compact. Thus, a trajectory $F^{+}x$ is divergent on average if and only if the trajectory $\unicode[STIX]{x1D70B}(F)\overline{\unicode[STIX]{x1D70B}}(x)$ is divergent on average, and consequently

(2.1)$$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D70B}}(\mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4}))=\mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4})). & & \displaystyle\end{eqnarray}$$

Let $\unicode[STIX]{x1D711}:\mathfrak{s}\rightarrow \mathfrak{g}$ be an embedding of Lie algebras such that $\text{d}\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D711}$ is the identity map. It follows from (2.1) that for any $x\in G/\unicode[STIX]{x1D6E4}$ and any $v\in \mathfrak{s}$

$$\begin{eqnarray}\exp (v)\overline{\unicode[STIX]{x1D70B}}(x)\in \mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4}))\end{eqnarray}$$

if and only if

$$\begin{eqnarray}\exp (\unicode[STIX]{x1D711}(v))\exp (v^{\prime })x\in \mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})\quad \text{for all }v^{\prime }\in \mathfrak{r}.\end{eqnarray}$$

For any $x\in G/\unicode[STIX]{x1D6E4}$, there is a neighborhood $U_{x}$ of $x$ of the form $U_{x}=\exp (V_{x})\exp (\unicode[STIX]{x1D711}(W_{x}))x$, where $V_{x}$ (respectively $W_{x}$) is an open neighborhood of $0$ in $\mathfrak{s}$ (respectively $\mathfrak{r}$) on which the exponential map is a bi-Lipschitz diffeomorphism onto its image. Here the metrics on $\mathfrak{s}$ and $\mathfrak{r}$ are given by Euclidean structures induced from that of $\mathfrak{g}$. Note that $\mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})\cap U_{x}$ is equal to

$$\begin{eqnarray}\{\exp (v)\exp (w)x:v\in V_{x},\exp (v)\overline{\unicode[STIX]{x1D70B}}(x)\in \mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4})),w\in W_{x}\}.\end{eqnarray}$$

Since the Hausdorff dimension is preserved by bi-Lipschitz maps and the induced metrics on homogeneous spaces are locally given by the metrics on Lie groups, we have

(2.2)$$\begin{eqnarray}\displaystyle \dim \mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})\cap U_{x} & = & \displaystyle \dim \{v\in V_{x}:\exp (v)\overline{\unicode[STIX]{x1D70B}}(x)\in \mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4}))\}\times W_{x}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \dim \{v\in V_{x}:\exp (v)\overline{\unicode[STIX]{x1D70B}}(x)\in \mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4}))\}+\dim \mathfrak{r}\nonumber\\ \displaystyle & = & \displaystyle \dim \mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4}))\cap \exp (V_{x})\overline{\unicode[STIX]{x1D70B}}(x)+\dim \mathfrak{r}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \dim \mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4}))+\dim \mathfrak{r},\end{eqnarray}$$

where Marstrand’s product theorem (see, e.g. [Reference Bishop and PeresBP17, Theorem 3.2.1]) is used to get the first inequality. Assuming that $\dim \mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4}))<\dim \mathfrak{s}$, then in view of (2.2) we have

$$\begin{eqnarray}\displaystyle \dim \mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4}) & = & \displaystyle \sup _{x\in G/\unicode[STIX]{x1D6E4}}\dim \mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})\cap U_{x}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \dim \mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4}))+\dim \mathfrak{r}\nonumber\\ \displaystyle & {<} & \displaystyle \dim \mathfrak{s}+\dim \mathfrak{r}=\dim G.\nonumber\end{eqnarray}$$

Hence to prove Theorem 1.2 it suffices to give a nontrivial upper bound of the Hausdorff dimension of $\mathfrak{D}(\unicode[STIX]{x1D70B}(F^{+}),S/\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6E4}))$. We summarize what we have obtained as follows.

Proof of Theorem 1.2 modulo Theorem 1.4.

According to Lemma 2.1, it suffices to prove the theorem under the additional assumption that $G$ is a center-free semisimple Lie group without compact factors. Under this assumption, the adjoint representation $\text{Ad}:G\rightarrow \text{SL}(\mathfrak{g})$ is a closed embedding. According to the real Jordan decomposition in Lemma 1.3, the compact part $K_{F}$ does not affect the divergence on average property of the trajectories. So we assume without loss of generality that $K_{F}$ is trivial.

There exist finitely many connected semisimple subgroups $G_{i}$ such that $G=\prod _{i}G_{i}$ and $\unicode[STIX]{x1D6E4}_{i}\,\stackrel{\text{def}}{=}\,\unicode[STIX]{x1D6E4}\cap G_{i}$ is an irreducible lattice of $G_{i}$ for each $i$. It follows that $\prod _{i}\unicode[STIX]{x1D6E4}_{i}$ is a finite-index subgroup of $\unicode[STIX]{x1D6E4}$ and the natural quotient map $G/\prod _{i}\unicode[STIX]{x1D6E4}_{i}\rightarrow G/\unicode[STIX]{x1D6E4}$ is proper. So we assume moreover that $\unicode[STIX]{x1D6E4}=\prod _{i}\unicode[STIX]{x1D6E4}_{i}$.

Denote by $\unicode[STIX]{x1D70B}_{j}$ the projection of $G$ to $G/G_{j}=\prod _{i\neq j}G_{i}$ and denote by $\overline{\unicode[STIX]{x1D70B}}_{j}$ the induced map from $G/\unicode[STIX]{x1D6E4}$ to $\unicode[STIX]{x1D70B}_{j}(G)/\unicode[STIX]{x1D70B}_{j}(\unicode[STIX]{x1D6E4})=\prod _{i\neq j}G_{i}/\unicode[STIX]{x1D6E4}_{i}$. Here if $G=G_{j}$ we interpret $\prod _{i\neq j}G_{i}$ as a trivial group and $\prod _{i\neq j}G_{i}/\unicode[STIX]{x1D6E4}_{i}$ as a single point set. If $G_{j}/\unicode[STIX]{x1D6E4}_{j}$ is compact or the projection of $A_{F}$ to $G_{j}$ is trivial, then

$$\begin{eqnarray}\mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})=\overline{\unicode[STIX]{x1D70B}}_{j}^{-1}\biggl(\mathfrak{D}\biggl(\unicode[STIX]{x1D70B}_{j}(F^{+}),\mathop{\prod }_{i\neq j}G_{i}/\unicode[STIX]{x1D6E4}_{i}\biggr)\biggr).\end{eqnarray}$$

So either $\mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})$ is an empty set or we finally can reduce the problem to the setting of Theorem 1.4 where each $\unicode[STIX]{x1D6E4}_{i}$ is a nonuniform lattice and the projection of $A_{F}$ to each $G_{i}$ is nontrivial. This completes the proof.◻

3 Preliminary on linear representations

From this section, we start the proof of Theorem 1.4. At the beginning of each section we will set up some notation that will be used later. Let $G$ and $F$ be as in Theorem 1.4. Let $A_{F}=\{a_{t}:t\in \mathbb{R}\}$ and $U_{F}=\{u_{t}:t\in \mathbb{R}\}$ be the diagonal and the unipotent parts of $F$, respectively. Let $H$ be the unique connected normal subgroup of $G$ such that $A_{F}\leqslant H$ and the projection of $A_{F}$ to each simple factor of $H$ is nontrivial. Since $A_{F}$ is nontrivial and $G$ is center-free, $H$ is the (nontrivial) product of some simple factors of $G$. Hence $H$ is a semisimple Lie group without compact factors. Let $S$ be the product of simple factors of $G$ not contained in $H$. Then $S$ is a semisimple and normal subgroup of $G$ that commutes with $H$. Moreover, $G=HS$ and $H\cap S=1_{G}$, where $1_{G}$ is the neutral element of $G$.

In this section we will prove a couple of auxiliary results for a finite-dimensional linear representation $\unicode[STIX]{x1D70C}:G\rightarrow \text{GL}(V)$ on a (nontrivial real) normed vector space $V$. These results will be used in the next section to prove a uniform contracting property of the EM height function. We will use $\Vert \cdot \Vert$ to denote the norm on $V$.

For $\unicode[STIX]{x1D706}\in \mathbb{R}$, we denote the $\unicode[STIX]{x1D706}$-Lyapunov subspace of $A_{F}$ by

$$\begin{eqnarray}V^{\unicode[STIX]{x1D706}}=\{v\in V:\unicode[STIX]{x1D70C}(a_{t})v=e^{\unicode[STIX]{x1D706}t}v\}.\end{eqnarray}$$

Recall that if $V^{\unicode[STIX]{x1D706}}\neq \{0\}$, then $\unicode[STIX]{x1D706}$ is a called an Lyapunov exponent of $(\unicode[STIX]{x1D70C},V)$. Since $U_{F}$ commutes with $A_{F}$, every Lyapunov subspace $V^{\unicode[STIX]{x1D706}}$ is $U_{F}$-invariant. As $A_{F}$ is $\mathbb{R}$-diagonalizable, the space $V$ can be decomposed as $V^{+}\oplus V^{0}\oplus V^{-}$ where

$$\begin{eqnarray}V^{+}=\bigoplus _{\unicode[STIX]{x1D706}>0}V^{\unicode[STIX]{x1D706}}\quad \text{and}\quad V^{-}=\bigoplus _{\unicode[STIX]{x1D706}<0}V^{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

Now we consider the adjoint representation of $G$ on the Lie algebra $\mathfrak{g}$ of $G$. It is easily checked that $\mathfrak{g}^{+},\mathfrak{g}^{-}$ and $\mathfrak{g}^{\text{c}}\,\stackrel{\text{def}}{=}\,\mathfrak{g}^{0}$ are subalgebras of $\mathfrak{g}$. The connected subgroup $G^{+}$ (respectively $G^{-}$) with Lie algebras $\mathfrak{g}^{+}$ (respectively $\mathfrak{g}^{-}$) is called unstable (respectively stable) horospherical subgroup of $a_{1}$. We denote the connected component of the centralizer of $a_{1}$ in $G$ by $G^{\text{c}}$ whose Lie algebra is $\mathfrak{g}^{\text{c}}$. Let $d,d^{\text{c}},d^{-}$ be the manifold dimensions of $G^{+}$, $G^{\text{c}}$ and $G^{-}$, respectively. It follows from the nontriviality of $A_{F}$ that $d>0$.

For $r\geqslant 0$ we let $B_{r}^{G}=\{h\in G:\operatorname{dist}(h,1_{G})<r\}$, $B_{r}^{\pm }=\{h\in G^{\pm }:\operatorname{dist}(h,1_{G})<r\}$ and $B_{r}^{\text{c}}=\{h\in G^{\text{c}}:\operatorname{dist}(h,1_{G})<r\}$. By rescaling the Riemannian metric if necessary, we may assume that:

1. (i) the product map $B_{1}^{-}\times B_{1}^{\text{c}}\times B_{1}^{+}\rightarrow G$ is a bi-Lipschitz diffeomorphism onto its image;

2. (ii) the logarithm map is well defined on $B_{1}^{G}$ and is a bi-Lipschitz diffeomorphism onto its image.

According to (i), it is safe to identify the direct product $B_{1}^{-}\times B_{1}^{\text{c}}\times B_{1}^{+}$ with its image $B_{1}^{-}B_{1}^{\text{c}}B_{1}^{+}\subset G$ and we will mainly use the latter notation for sake of convenience. The same statement as (ii) also holds for $B_{1}^{\pm }$ and $B_{1}^{\text{c}}$.

We fix a Haar measure $\unicode[STIX]{x1D707}$ on $G^{+}$ normalized with $\unicode[STIX]{x1D707}(B_{1}^{+})=1$. Since the metric ‘$\operatorname{dist}$’ is right invariant, any open ball of radius $r$ in $G^{+}$ has the form $B_{r}^{+}h$$(h\in G^{+})$ and there exists $C_{0}\geqslant 1$ such that

(3.1)$$\begin{eqnarray}\displaystyle C_{0}^{-1}r^{d}\leqslant \unicode[STIX]{x1D707}(B_{r}^{+}h)=\unicode[STIX]{x1D707}(B_{r}^{+})\leqslant C_{0}r^{d}\quad \text{for all }0\leqslant r\leqslant 1. & & \displaystyle\end{eqnarray}$$

For $g,h\in G$ we let $g^{h}=h^{-1}gh$. For $z\in G^{\text{c}}$, let $F_{z}=\{f_{t}^{z}:t\in \mathbb{R}\}$ and $F_{z}^{+}=\{f_{t}^{z}:t\geqslant 0\}$. Note that $f_{t}^{z}=a_{t}u_{t}^{z}$ and

(3.2)$$\begin{eqnarray}\displaystyle \{u_{1}^{z}:z\in B_{1}^{\text{c}}\}\quad \text{is relatively compact.} & & \displaystyle\end{eqnarray}$$

Lemma 3.1. Let $\unicode[STIX]{x1D70C}:G\rightarrow \text{GL}(V)$ be a representation on a finite-dimensional normed vector space $V$. Let $\unicode[STIX]{x1D706}$ be a Lyapunov exponent of $(\unicode[STIX]{x1D70C},V)$. For any $\unicode[STIX]{x1D6FF}>0$, there exists $T_{\unicode[STIX]{x1D6FF}}>0$ such that, for all $t\geqslant T_{\unicode[STIX]{x1D6FF}},z\in B_{1}^{\text{c}}$ and unit vector $v\in V^{\unicode[STIX]{x1D706}}$ we have

(3.3)$$\begin{eqnarray}e^{(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D6FF})t}\leqslant \Vert \unicode[STIX]{x1D70C}(f_{t}^{z})v\Vert \leqslant e^{(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})t}.\end{eqnarray}$$

Proof. For all $v\in V^{\unicode[STIX]{x1D706}}$ with $\Vert v\Vert =1$ we have $\Vert \unicode[STIX]{x1D70C}(a_{t})v\Vert =e^{\unicode[STIX]{x1D706}t}$. On the other hand, in view of (3.2), there exist $C>0$ and $n\in \mathbb{N}$ (in this paper $\mathbb{N}=\{1,2,3,\ldots \}$) such that

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D70C}(u_{t}^{z})\Vert \leqslant C(|t|+1)^{n}\end{eqnarray}$$

for all $z\in B_{1}^{\text{c}}$ and $t\in \mathbb{R}$. Therefore, for any unit vector $v\in V^{\unicode[STIX]{x1D706}},z\in B_{1}^{\text{c}}$ and sufficiently large $t$,

$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}(f_{t}^{z})v\Vert & {\geqslant} & \displaystyle \Vert \unicode[STIX]{x1D70C}(u_{-t}^{z})\Vert ^{-1}\Vert \unicode[STIX]{x1D70C}(a_{t})v\Vert \geqslant C^{-1}(|t|+1)^{-n}e^{\unicode[STIX]{x1D706}t}\geqslant e^{(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D6FF})t},\nonumber\\ \displaystyle \Vert \unicode[STIX]{x1D70C}(f_{t}^{z})v\Vert & {\leqslant} & \displaystyle \Vert \unicode[STIX]{x1D70C}(u_{t}^{z})\Vert \Vert \unicode[STIX]{x1D70C}(a_{t})v\Vert \leqslant C(|t|+1)^{n}e^{\unicode[STIX]{x1D706}t}\leqslant e^{(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})t}.\Box \nonumber\end{eqnarray}$$

Since $z\in G^{\text{c}}$, it follows from the above discussion that the adjoint action of $f_{t}^{z}$ preserves each Lyapunov subspaces, and hence it preserves $\mathfrak{g}^{+},\mathfrak{g}^{-}$ and $\mathfrak{g}^{\text{c}}$. Consequently, the conjugate action of $f_{t}^{z}$ preserves $G^{+},G^{-},G^{\text{c}}$. Moreover, we have the following.

Lemma 3.2. Let $\unicode[STIX]{x1D706}$ be the top Lyapunov exponent of $A_{F}$ in the representation $(\text{Ad},\mathfrak{g})$, i.e. $\unicode[STIX]{x1D706}=\max \{r\in \mathbb{R}:\mathfrak{g}^{r}\neq \{0\}\}$. Then for any $\unicode[STIX]{x1D6FF}>0$, there exists $T_{\unicode[STIX]{x1D6FF}}^{\prime }>0$ such that, for all $t\geqslant T_{\unicode[STIX]{x1D6FF}}^{\prime }$ and $z\in B_{1}^{\text{c}}$, we have

(3.4)$$\begin{eqnarray}\displaystyle B_{e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})t}r}^{+} & \subset & \displaystyle f_{-t}^{z}B_{r}^{+}f_{t}^{z}\subset B_{r/4}^{+}\quad \text{for all }r\leqslant 1,\end{eqnarray}$$

(3.5)$$\begin{eqnarray}\displaystyle B_{e^{-\unicode[STIX]{x1D6FF}t}r}^{\text{c}} & \subset & \displaystyle f_{-t}^{z}B_{r}^{\text{c}}f_{t}^{z}\subset B_{e^{\unicode[STIX]{x1D6FF}t}r}^{\text{c}}\quad \text{for all }r\leqslant e^{-\unicode[STIX]{x1D6FF}t}.\end{eqnarray}$$

Proof. We only give detailed proofs for the second inclusion of (3.5). The proofs of the remaining inclusions are similar, and hence are omitted.

By assumption, the logarithm map from $B_{1}^{G}$ to $\mathfrak{g}$ is a bi-Lipschitz diffeomorphism onto its image. Hence there exists $R\geqslant 1$ such that, for any $h_{1},h_{2}\in B_{1}^{G}$, we have

$$\begin{eqnarray}R^{-1}\operatorname{dist}(\log h_{1},\log h_{2})\leqslant \operatorname{dist}(h_{1},h_{2})\leqslant R\operatorname{dist}(\log h_{1},\log h_{2}).\end{eqnarray}$$

By applying Lemma 3.1 to the adjoint representation of $G$, we know that there exists $T_{\unicode[STIX]{x1D6FF}/2}>2\log R$ such that, for all $t\geqslant T_{\unicode[STIX]{x1D6FF}/2},v\in \mathfrak{g}^{\text{c}}$ and $z\in B_{1}^{\text{c}}$, we have

$$\begin{eqnarray}\Vert \text{Ad}(f_{-t}^{z})v\Vert \leqslant e^{\unicode[STIX]{x1D6FF}t/2}\Vert v\Vert .\end{eqnarray}$$

By taking $T_{\unicode[STIX]{x1D6FF}}^{\prime }\geqslant T_{\unicode[STIX]{x1D6FF}/2}$ large enough, we may assume that $\text{Ad}(f_{-t}^{z})v\in \log B_{1}^{\text{c}}$ for all $t\geqslant T_{\unicode[STIX]{x1D6FF}}^{\prime }$ and $v\in \log B_{r}^{\text{c}}$ where $r\leqslant e^{-\unicode[STIX]{x1D6FF}t}$. Thus for $r\leqslant e^{-\unicode[STIX]{x1D6FF}t}$ and $h\in B_{r}^{\text{c}}$

$$\begin{eqnarray}\displaystyle \operatorname{dist}(f_{-t}^{z}hf_{t}^{z},1_{G}) & = & \displaystyle \operatorname{dist}(f_{-t}^{z}\exp (\log h)f_{t}^{z},1_{G})=\operatorname{dist}(\exp (\text{Ad}(f_{-t}^{z})\log h),1_{G})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle R\operatorname{dist}(\text{Ad}(f_{-t}^{z})\log h,0)\leqslant Re^{\unicode[STIX]{x1D6FF}t/2}\operatorname{dist}(\log h,0)\leqslant R^{2}e^{\unicode[STIX]{x1D6FF}t/2}\operatorname{dist}(h,1_{G})<e^{\unicode[STIX]{x1D6FF}t}r.\nonumber\end{eqnarray}$$

This completes the proof of the second inclusion in (3.5). ◻

From now on till the end of this section, we assume that $\unicode[STIX]{x1D70C}:G\rightarrow \text{GL}(V)$ is a representation on a finite-dimensional normed vector space $V$ which has no nonzero $H$-invariant vectors. As any two norms on $V$ are equivalent, we also assume without loss of generality that the norm is Euclidean.

Recall that a nonzero $H$-invariant subspace $V^{\prime }$ of $V$ is said to be $H$-irreducible if $V^{\prime }$ contains no $H$-invariant subspaces besides $\{0\}$ and itself. The complete reducibility of representations of $H$ implies that there exists a unique decomposition (called $H$-isotropic decomposition)

(3.6)$$\begin{eqnarray}\displaystyle V=V_{1}\oplus \cdots \oplus V_{m} & & \displaystyle\end{eqnarray}$$

such that irreducible sub-representations of $H$ in the same $V_{i}$ are isomorphic but irreducible sub-representations in different $V_{i}$ are nonisomorphic. Since $S$ commutes with $H$, each $V_{i}$ is $S$-invariant, and hence $G$-invariant. Each $V_{i}$ is called an $H$-isotropic subspace of $V$.

Let $\unicode[STIX]{x1D706}_{i}$ be the top Lyapunov exponent of $A_{F}$ in $(\unicode[STIX]{x1D70C},V_{i})$, i.e.,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{i}=\max \{\unicode[STIX]{x1D706}\in \mathbb{R}:V_{i}^{\unicode[STIX]{x1D706}}\neq \{0\}\}. & & \displaystyle \nonumber\end{eqnarray}$$

Since the projection of $A_{F}$ to each simple factor of $H$ is nontrivial, every $\unicode[STIX]{x1D706}_{i}$ is positive. Let $\unicode[STIX]{x1D706}$ be the minimum of top Lyapunov exponents in all the $V_{i}$, i.e.,

(3.7)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}=\min \{\unicode[STIX]{x1D706}_{i}:1\leqslant i\leqslant m\}>0. & & \displaystyle\end{eqnarray}$$

Let $\unicode[STIX]{x1D70B}_{i}:V_{i}\rightarrow V_{i}^{\unicode[STIX]{x1D706}_{i}}$ be the $A_{F}$-equivariant projection.

Lemma 3.3. For all $v\in V_{i}\smallsetminus \{0\}$, the map

(3.8)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{v}:G^{+}\mapsto \mathbb{R}\quad \text{where }\unicode[STIX]{x1D711}_{v}(h)=\Vert \unicode[STIX]{x1D70B}_{i}(\unicode[STIX]{x1D70C}(h)v)\Vert ^{2} & & \displaystyle\end{eqnarray}$$

is not identically zero.

Lemma 3.4. For all $v\in V_{i}\smallsetminus \{0\}$ and $r\geqslant 0$, let

$$\begin{eqnarray}E(v,r)=\{h\in B_{1}^{+}:\Vert \unicode[STIX]{x1D70B}_{i}(\unicode[STIX]{x1D70C}(h)v)\Vert \leqslant r\}.\end{eqnarray}$$

Then there exists $\unicode[STIX]{x1D703}_{i}>0$ such that

(3.9)$$\begin{eqnarray}C_{i}\,\stackrel{\text{def}}{=}\,\sup _{\substack{ \Vert v\Vert =1,v\in V_{i} \\ r>0}}r^{-\unicode[STIX]{x1D703}_{i}}\unicode[STIX]{x1D707}(E(v,r))<\infty .\end{eqnarray}$$

In particular, $\unicode[STIX]{x1D707}(E(v,0))=0$.

Proof. Since $G^{+}$ is a unipotent group, it is simply connected and by [Reference Corwin and GreenleafCG90, Theorem 1.2.10(a)] there is an isomorphism of affine varieties $\mathbb{R}^{d}\rightarrow G^{+}$ such that the Lebesgue measure of $\mathbb{R}^{d}$ corresponds to the Haar measure $\unicode[STIX]{x1D707}$. During the proof, we will identify the group $G^{+}$ with $\mathbb{R}^{d}$ for convenience.

By Lemma 3.3, for every nonzero $v\in V_{i}$ the map $\unicode[STIX]{x1D711}_{v}$ in (3.8) is a nonzero polynomial map. So $\unicode[STIX]{x1D711}_{v}|_{B_{1}^{+}}$ is nonzero. Note that the degrees of $\unicode[STIX]{x1D711}_{v}$ ($v\in V_{i}$) are uniformly bounded from above. Therefore, the $(C,\unicode[STIX]{x1D6FC})$-good property of polynomials in [Reference Bernik, Kleinbock and MargulisBKM01, §3] implies that there exist positive constants $C$ and $\unicode[STIX]{x1D6FC}$ such that

(3.10)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(E(v,r))\leqslant C\biggl(\frac{r^{2}}{\sup _{h\in B_{1}^{+}}\unicode[STIX]{x1D711}_{v}(h)}\biggr)^{\unicode[STIX]{x1D6FC}} & & \displaystyle\end{eqnarray}$$

for all nonzero $v\in V_{i}$. Since the set of unit vectors of $V_{i}$ is compact,

(3.11)$$\begin{eqnarray}\displaystyle \inf _{\Vert v\Vert =1,v\in V_{i}}\sup _{h\in B_{1}^{+}}\unicode[STIX]{x1D711}_{v}(h)>0. & & \displaystyle\end{eqnarray}$$

So (3.9) follows from (3.10) and (3.11) by taking $\unicode[STIX]{x1D703}_{i}=2\unicode[STIX]{x1D6FC}$.◻

Lemma 3.6. Let $\unicode[STIX]{x1D703}_{0}=\min _{1\leqslant i\leqslant m}\unicode[STIX]{x1D703}_{i}$ where $\unicode[STIX]{x1D703}_{i}>0$ so that Lemma 3.4 holds and let $\unicode[STIX]{x1D706}$ be as in (3.7). Then for any $0<\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D703}<\unicode[STIX]{x1D703}_{0}$, there exists $T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}>0$ such that, for all $t\geqslant T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}$, $z\in B_{1}^{\text{c}}$ and $v\in V$ with $\Vert v\Vert =1$, we have

(3.12)$$\begin{eqnarray}\int _{B_{1}^{+}}\Vert \unicode[STIX]{x1D70C}(f_{t}^{z}h)v\Vert ^{-\unicode[STIX]{x1D703}}\,d\unicode[STIX]{x1D707}(h)\leqslant e^{-(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D706}t}.\end{eqnarray}$$

Proof. Without loss of generality, we assume further that the Euclidean norm $\Vert \cdot \Vert$ on $V$ satisfies the following properties:

1. – the Lyapunov subspaces of $A_{F}$ are orthogonal to each other;

2. – the $H$-isotropic subspaces $V_{i}~(1\leqslant i\leqslant m)$ are orthogonal to each other.

Let

(3.13)$$\begin{eqnarray}R_{i}=\sup _{v\in V_{i},\Vert v\Vert =1,h\in B_{1}^{+}}\Vert \unicode[STIX]{x1D70B}_{i}(\unicode[STIX]{x1D70C}(h)v)\Vert \quad \text{and}\quad R=\max \{R_{i}:1\leqslant i\leqslant m\}.\end{eqnarray}$$

Let $C=\max \{C_{i}:1\leqslant i\leqslant m\}$ where $C_{i}$ is given in (3.9). Let $\unicode[STIX]{x1D703}^{\prime }=\max \{\unicode[STIX]{x1D703}_{i}:1\leqslant i\leqslant m\}$.

According to Lemma 3.1, there exists $T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}^{\prime \prime }>0$ such that (3.3) holds for any $t\geqslant T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}^{\prime \prime }$, any nonzero $v\in V^{\unicode[STIX]{x1D706}_{i}}~(1\leqslant i\leqslant m)$ and any $z\in B_{1}^{\text{c}}$, i.e.,

$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}(f_{t}^{z})v\Vert ^{-\unicode[STIX]{x1D703}}\leqslant e^{-(1-\unicode[STIX]{x1D6FF}/2\unicode[STIX]{x1D703})\unicode[STIX]{x1D703}\unicode[STIX]{x1D706}_{i}t}\Vert v\Vert ^{-\unicode[STIX]{x1D703}}\leqslant e^{-(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D706}t}\Vert v\Vert ^{-\unicode[STIX]{x1D703}}. & & \displaystyle \nonumber\end{eqnarray}$$

This inequality and the assumption of the norm implies that for all nonzero $v\in V_{i}$ and $t\geqslant T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}^{\prime \prime }$

(3.14)$$\begin{eqnarray}\Vert \unicode[STIX]{x1D70C}(f_{t}^{z}h)v\Vert ^{-\unicode[STIX]{x1D703}}\leqslant e^{-(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D706}t}\Vert \unicode[STIX]{x1D70B}_{i}(\unicode[STIX]{x1D70C}(h)v)\Vert ^{-\unicode[STIX]{x1D703}},\end{eqnarray}$$

where ${\textstyle \frac{1}{0}}$ is interpreted as $\infty$. Let $T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}\geqslant T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}^{\prime \prime }$ be a large enough real number so that $t\geqslant T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}$ implies

(3.15)$$\begin{eqnarray}\frac{(2m)^{\unicode[STIX]{x1D703}^{\prime }}CR^{\unicode[STIX]{x1D703}^{\prime }-\unicode[STIX]{x1D703}}}{1-2^{\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{0}}}e^{-(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D706}t}\leqslant e^{-(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D706}t}.\end{eqnarray}$$

Let $v$ be a unit vector of $V$. We write $v=v_{1}+\cdots +v_{m}$ where $v_{i}\in V_{i}$. Since we assume different $V_{i}$ are orthogonal to each other, there exists an integer $i\in [1,m]$ such that $m\Vert v_{i}\Vert \geqslant \Vert v\Vert =1$.

There is a disjoint union decomposition of $B_{1}^{+}$ as

$$\begin{eqnarray}E(v_{i},0)\cup \biggl(\mathop{\bigcup }_{n\geqslant 0}E^{+}(v_{i},2^{-n}R_{i})\biggr),\end{eqnarray}$$

where

$$\begin{eqnarray}E^{+}(v_{i},2^{-n}R_{i})=E(v_{i},2^{-n}R_{i})\smallsetminus E(v_{i},2^{-n-1}R_{i}).\end{eqnarray}$$

Since $\unicode[STIX]{x1D707}(E(v_{i},0))=0$, for any $z\in B_{1}^{\text{c}}$ and $t\geqslant T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}$ we have

4 Eskin–Margulis height function

Let the notation be as in Theorem 1.4. In this section, we will establish a uniform contraction property of the EM height function on $G/\unicode[STIX]{x1D6E4}$ with respect to a family of one-parameter subgroups $F_{z}~(z\in B_{1}^{\text{c}})$.

Recall that $G/\unicode[STIX]{x1D6E4}=\prod _{i=1}^{m}G_{i}/\unicode[STIX]{x1D6E4}_{i}$ where each $G_{i}/\unicode[STIX]{x1D6E4}_{i}$ is a noncompact irreducible quotient of a semisimple Lie group without compact factors. Since we assume the projection of $A_{F}$ to each $G_{i}$ is nontrivial, we have $H=\prod _{i=1}^{m}H_{i}$, where each $H_{i}=G_{i}\cap H$ is a nontrivial connected normal subgroup of $G_{i}$.

Let us recall the definition of the EM height function from [Reference Eskin and MargulisEM04]. The EM height function is constructed on each $G_{i}/\unicode[STIX]{x1D6E4}_{i}$ using a finite set $\unicode[STIX]{x1D6E5}_{i}$ of $\unicode[STIX]{x1D6E4}_{i}$-rational parabolic subgroups of $G_{i}$. Recall that a parabolic subgroup $P$ of $G_{i}$ is $\unicode[STIX]{x1D6E4}_{i}$-rational if the unipotent radical of $P$ intersects $\unicode[STIX]{x1D6E4}_{i}$ in a lattice. If the rank of $G_{i}$ is bigger than one, then Margulis’ arithmeticity theorem implies that there is a $\mathbb{Q}$-structure on $G_{i}$ such that $\unicode[STIX]{x1D6E4}_{i}$ is commensurable with $G_{i}(\mathbb{Z})$. In this case the set $\unicode[STIX]{x1D6E5}_{i}$ consists of standard $\mathbb{Q}$-rational maximal parabolic subgroups of $G_{i}$ with respect to a fixed $\mathbb{Q}$-split torus and fixed positive roots. So the irreducibility of $\unicode[STIX]{x1D6E4}_{i}$ implies that no conjugates of $H_{i}$ is contained in any $P\in \unicode[STIX]{x1D6E5}_{i}$. The same conclusion holds in the case where $G_{i}$ has rank one. The reason is that in this case $H_{i}=G_{i}$ and $\unicode[STIX]{x1D6E5}_{i}=\{P\}$ where $P$ is a maximal parabolic subgroup defined over $\mathbb{R}$.

For each $P_{i,j}\in \unicode[STIX]{x1D6E5}_{i}$, there exists a representation $\unicode[STIX]{x1D70C}_{i,j}:G_{i}\rightarrow \text{GL}(V_{i,j})$ on a normed vector space and a nonzero vector $w_{i,j}\in V_{i,j}$ such that the stabilizer of $\mathbb{R}w_{i,j}$ is $P_{i,j}$. In fact, the map $\unicode[STIX]{x1D70C}_{i,j}$ is given by Chevalley’s theorem and is defined over $\mathbb{Q}$ when $\unicode[STIX]{x1D6E4}$ is arithmetic. We consider $\unicode[STIX]{x1D70C}_{i,j}$ as a representation of $G$ so that $\unicode[STIX]{x1D70C}(G_{s})$ is the identity linear map if $s\neq i$. Let $V_{i,j}^{H}$ be the $H$-invariant subspace of $V_{i,j}$ consisting of $H$-invariant vectors. Let $\unicode[STIX]{x1D70B}_{i,j}$ be the projection of $V_{i,j}$ to the $H$-invariant subspace $V_{i,j}^{\prime }$ complementary to $V_{i,j}^{H}$. Since no conjugates of $H_{i}$ is contained in $P_{i,j}$ and $G_{i}=K_{i}P_{i,j}$ for some maximal compact subgroup $K_{i}$ of $G_{i}$, there exists $C\geqslant 1$ such that

$$\begin{eqnarray}\Vert v\Vert \leqslant C\Vert \unicode[STIX]{x1D70B}_{i,j}(v)\Vert\end{eqnarray}$$

for all $v\in \unicode[STIX]{x1D70C}_{i,j}(G)w_{i,j}$. Note that $V_{i,j}^{\prime }$ is $G$-invariant and it has no nonzero $H$-invariant vectors. Therefore, Lemma 3.6 implies the following lemma which corresponds to Condition A in [Reference Eskin and MargulisEM04]. For simplicity, the set of indices $j$ such that $P_{i,j}\in \unicode[STIX]{x1D6E5}_{i}$ is also denoted as $\unicode[STIX]{x1D6E5}_{i}$. This will not cause any confusion.

Lemma 4.1. For any $1\leqslant i\leqslant m$ and $j\in \unicode[STIX]{x1D6E5}_{i}$, there exist positive constants $\unicode[STIX]{x1D703}_{0}^{i,j}$ and $\unicode[STIX]{x1D706}^{i,j}$ such that, for any $0<\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D703}<\unicode[STIX]{x1D703}_{0}^{i,j}$, any nonzero $v\in \unicode[STIX]{x1D70C}_{i,j}(G)w_{i,j}$ and any $z\in B_{1}^{\text{c}}$, one has

(4.1)$$\begin{eqnarray}\displaystyle \int _{B_{1}^{+}}\Vert \unicode[STIX]{x1D70C}_{i,j}(f_{t}^{z}h)v\Vert ^{-\unicode[STIX]{x1D703}}\,dh\leqslant e^{-(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FF})t\unicode[STIX]{x1D706}^{i,j}}\Vert v\Vert ^{-\unicode[STIX]{x1D703}} & & \displaystyle\end{eqnarray}$$

provided $t\geqslant T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}^{i,j}$ where $T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}^{i,j}>0$ is a constant depending on $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6FF}$.

Proof. We assume without loss of generality that for all $V_{i,j}$ the norm $\Vert \cdot \Vert$ is Euclidean and $V_{i,j}^{H}$ and $V_{i,j}^{\prime }$ are orthogonal to each other. According to Lemma 3.6, for each representation $\unicode[STIX]{x1D70C}_{i,j}|_{V_{i,j}^{\prime }}$, there exist positive constants $\unicode[STIX]{x1D703}_{0}^{i,j}$ and $\unicode[STIX]{x1D706}^{i,j}$ with the following properties: for any $0<\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D703}<\unicode[STIX]{x1D703}_{0}^{i,j}$ there exists $T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}}>0$ such that, for any $t\geqslant T_{\unicode[STIX]{x1D703},\unicode[STIX]{x1D6FF}},z\in B_{1}^{\text{c}}$ and any nonzero $v\in \unicode[STIX]{x1D70C}_{i,j}(G)w_{i,j}$, one has

$$\begin{eqnarray}\displaystyle \int _{B_{1}^{+}}\Vert \unicode[STIX]{x1D70C}_{i,j}(f_{t}^{z}h)v\Vert ^{-\unicode[STIX]{x1D703}}\,dh & {\leqslant} & \displaystyle \int _{B_{1}^{+}}\Vert \unicode[STIX]{x1D70C}_{i,j}(f_{t}^{z}h)\unicode[STIX]{x1D70B}_{i,j}(v)\Vert ^{-\unicode[STIX]{x1D703}}\,dh\nonumber\\ \displaystyle & {\leqslant} & \displaystyle e^{-(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FF})t\unicode[STIX]{x1D706}^{i,j}}\Vert \unicode[STIX]{x1D70B}_{i,j}(v)\Vert ^{-\unicode[STIX]{x1D703}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle C^{\unicode[STIX]{x1D703}}e^{-(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FF})t\unicode[STIX]{x1D706}^{i,j}}\Vert v\Vert ^{-\unicode[STIX]{x1D703}}.\nonumber\end{eqnarray}$$

It is not hard to see from the above estimate that (4.1) holds for sufficiently large $t$.◻

Besides $\unicode[STIX]{x1D70C}_{i,j}$, the EM height function is constructed using positive constants $c_{i,j}$ and $q_{i,j}$ which are combinatorial data determined by the root system, see [Reference Eskin and MargulisEM04, (3.22), (3.28)]. Although only the product $c_{i,j}q_{i,j}$ is used in this paper, the constants $c_{i,j}$ and $q_{i,j}$ are given by different combinatorial data and we use both of them for the consistency with [Reference Eskin and MargulisEM04]. We assume the norm $\Vert \cdot \Vert$ on $V_{i,j}$ is $K_{i}$-invariant and let

(4.2)$$\begin{eqnarray}\displaystyle u_{i,j}(g\unicode[STIX]{x1D6E4})=\max _{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}}\frac{1}{\Vert \unicode[STIX]{x1D70C}_{i,j}(g\unicode[STIX]{x1D6FE})w_{i,j}\Vert ^{1/c_{i,j}q_{i,j}}}, & & \displaystyle\end{eqnarray}$$

where $g\in G$. The maximum is used in the definition of $u_{i,j}$ since supremum on the right-hand side is achieved for some $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$. In the case where $\unicode[STIX]{x1D6E4}$ is arithmetic this follows from $\mathbb{Q}$-rationality of $\unicode[STIX]{x1D70C}_{i,j}$. The rank one case is also well known, see e.g. [Reference Kleinbock and WeissKW13, Proposition 3.1]. The main property of $u_{i,j}$ proved in [Reference Eskin and MargulisEM04, §3.2] is summarized in the following lemma.

The values $\unicode[STIX]{x1D706}_{i,jk}$ are given in [Reference Eskin and MargulisEM04, (3.31)] and (i) is [Reference Eskin and MargulisEM04, (3.32)]. The statement of Lemma 4.2(ii) is summarized from the proof of [Reference Eskin and MargulisEM04, (3.35)] and (4.3) is the explicit version of [Reference Eskin and MargulisEM04, (3.35)]. We can make the values $C^{\prime }$ and $b^{\prime }$ independent of $i$ since there are only finitely many $i$. The dependence of $C^{\prime }$ and $b^{\prime }$ on $\unicode[STIX]{x1D703}$ can be removed if $\unicode[STIX]{x1D703}$ is bounded from above.

Let

(4.4)$$\begin{eqnarray}\unicode[STIX]{x1D703}_{1}=\max \bigg\{\unicode[STIX]{x1D703}>0:\frac{\unicode[STIX]{x1D703}}{c_{i,j}q_{ij}}\leqslant \unicode[STIX]{x1D703}_{0}^{i,j}\text{ for all }i,j\bigg\}\quad \text{and}\quad \unicode[STIX]{x1D6FC}_{1}=\min _{i,j}\bigg\{\frac{\unicode[STIX]{x1D703}_{1}}{c_{i,j}q_{ij}}\unicode[STIX]{x1D706}^{i,j}\bigg\},\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{0}^{i,j}$ and $\unicode[STIX]{x1D706}^{i,j}$ are constants given by Lemma 4.1. We call $\unicode[STIX]{x1D6FC}_{1}$ a contraction rate of the dynamical system $(G/\unicode[STIX]{x1D6E4},F^{+})$.

Lemma 4.4. For every $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{1}$, there exist $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}(\unicode[STIX]{x1D6FC})\in (0,\unicode[STIX]{x1D703}_{1})$ and $T=T(\unicode[STIX]{x1D6FC})>0$ such that, for all $t\geqslant T$ and $\unicode[STIX]{x1D716}$ sufficiently small depending on $t$, the EM height function

(4.5)$$\begin{eqnarray}\displaystyle u:G/\unicode[STIX]{x1D6E4}\rightarrow (0,\infty )\quad \text{defined by }u(x)=\mathop{\sum }_{i\in [1,m],j\in \unicode[STIX]{x1D6E5}_{i}}(\unicode[STIX]{x1D716}\,u_{i,j}(x))^{\unicode[STIX]{x1D703}} & & \displaystyle\end{eqnarray}$$

satisfies the following properties:

1. (i) $u(x)\rightarrow \infty$ if and only if $x\rightarrow \infty$ in $G/\unicode[STIX]{x1D6E4}$;

2. (ii) for any compact subset $K$ of $G$, there exists $C=C(K,t)\geqslant 1$ such that $u(hx)\leqslant Cu(x)$ for all $h\in K$ and $x\in G/\unicode[STIX]{x1D6E4}$;

3. (iii) there exists $b=b(t)>0$ such that, for all $z\in B_{1}^{\text{c}}$ and $x\in G/\unicode[STIX]{x1D6E4}$, one has

   (4.6)$$\begin{eqnarray}\displaystyle \int _{B_{1}^{+}}u(f_{t}^{z}hx)\,d\unicode[STIX]{x1D707}(h)<\frac{1}{2}e^{-\unicode[STIX]{x1D6FC}t}u(x)+b; & & \displaystyle\end{eqnarray}$$

4. (iv) there exists $\ell =\ell (t)\geqslant 1$ such that if $u(x)\geqslant \ell$, then for all $z\in B_{1}^{\text{c}}$

   (4.7)$$\begin{eqnarray}\displaystyle \int _{B_{1}^{+}}u(f_{t}^{z}hx)\,d\unicode[STIX]{x1D707}(h)<e^{-\unicode[STIX]{x1D6FC}t}u(x). & & \displaystyle\end{eqnarray}$$

The proof of Lemma 4.4 is essentially the same as in [Reference Eskin and MargulisEM04, §3.2], with the exception of the following minor differences: (i) we do not assume that $\unicode[STIX]{x1D6E4}$ is irreducible; (ii) we want (4.6) and (4.7) holds uniformly in $z$; (iii) we want to make the contraction constant in (4.6) as explicit as $\frac{1}{2}e^{-\unicode[STIX]{x1D6FC}t}$. A sketch of proof will be given below to show how to adapt the proof given in [Reference Eskin and MargulisEM04, §3.2] to our settings.

Sketch of proof.

It follows from the corresponding properties for each function $u_{i,j}$ proved in [Reference Eskin and MargulisEM04, §3.2] that the first two conclusions hold for any choice of $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D716}$. Note that (iv) is a direct corollary of (iii).

Now we prove (iii). We fix $\unicode[STIX]{x1D6FF}>0$ sufficiently small such that

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}+\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}^{i,j}}{c_{i,j}q_{i,j}}<\unicode[STIX]{x1D6FC}_{1}\quad \forall i,j.\end{eqnarray}$$

According to the definitions of $\unicode[STIX]{x1D703}_{1}$, $\unicode[STIX]{x1D6FC}_{1}$ and the choice of $\unicode[STIX]{x1D6FF}$ above, there exists $\unicode[STIX]{x1D703}>0$ such that

(4.8)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}<\unicode[STIX]{x1D703}_{1}\quad \text{and}\quad \frac{(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D706}^{i,j}}{c_{i,j}q_{i,j}}\geqslant \unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}\quad \forall i,j. & & \displaystyle\end{eqnarray}$$

Let $\unicode[STIX]{x1D6FF}_{i,j}=\unicode[STIX]{x1D6FF}/c_{i,j}q_{i,j}$, $\unicode[STIX]{x1D703}_{i,j}=\unicode[STIX]{x1D703}/c_{i,j}q_{i,j}$, then according to Lemma 4.1 there exists $T^{i,j}>0$ such that, for $t\geqslant T^{i,j}$, one has that (4.1) holds with $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{i,j}$ and $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}_{i,j}$. We will show that Lemma 4.4 holds for $T=\unicode[STIX]{x1D6FF}^{-1}\log 4+\max _{i,j}T^{i,j}$.

Now let us fix $z\in B_{1}^{\text{c}}$ and $x=g\unicode[STIX]{x1D6E4}\in G/\unicode[STIX]{x1D6E4},t\geqslant T$ and an index $i,j$. By (4.8), Lemmas 4.1 and 4.2 we have

(4.9)$$\begin{eqnarray}\int _{B_{1}^{+}}u_{i,j}^{\unicode[STIX]{x1D703}}(f_{t}^{z}hx)\,d\unicode[STIX]{x1D707}(h)\leqslant e^{-(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF})t}u_{i,j}^{\unicode[STIX]{x1D703}}(x)+C^{\prime }\mathop{\prod }_{k\neq j}u_{i,k}^{\unicode[STIX]{x1D703}\unicode[STIX]{x1D706}_{i,jk}}(x)+b^{\prime },\end{eqnarray}$$

where $C^{\prime }$ and $b^{\prime }$ are positive constants depending on $t$.

Now we fix $\unicode[STIX]{x1D716}>0$ sufficiently small which will be specified later and define $u(x)$ as in (4.5). We can write (4.9) as

$$\begin{eqnarray}\int _{B_{1}^{+}}\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D703}}u_{i,j}^{\unicode[STIX]{x1D703}}(f_{t}^{z}hx)\,d\unicode[STIX]{x1D707}(h)\leqslant e^{-(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF})t}\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D703}}u_{i,j}^{\unicode[STIX]{x1D703}}(x)+\unicode[STIX]{x1D716}_{i,j}\mathop{\prod }_{k\neq j}\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D703}\unicode[STIX]{x1D706}_{i,jk}}u_{i,k}^{\unicode[STIX]{x1D703}\unicode[STIX]{x1D706}_{i,jk}}(x)+\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D703}}b^{\prime },\end{eqnarray}$$

where $\unicode[STIX]{x1D716}_{i,j}=C^{\prime }\unicode[STIX]{x1D716}^{1-\sum _{k\neq j}\unicode[STIX]{x1D706}_{i,jk}}$. According to Lemma 4.2(1) and the Jensen’s inequality, we have

$$\begin{eqnarray}\mathop{\prod }_{k\neq j}(\unicode[STIX]{x1D716}u_{i,k}(x))^{\unicode[STIX]{x1D703}\unicode[STIX]{x1D706}_{i,jk}}\leqslant \mathop{\prod }_{\substack{ k\neq j \\ \unicode[STIX]{x1D716}u_{i,k}(x)\geqslant 1}}(\unicode[STIX]{x1D716}u_{i,k}(x))^{\unicode[STIX]{x1D703}\unicode[STIX]{x1D706}_{i,jk}}\leqslant 1+\mathop{\sum }_{k\neq j}(\unicode[STIX]{x1D716}u_{i,k}(x))^{\unicode[STIX]{x1D703}}.\end{eqnarray}$$

Hence we have

$$\begin{eqnarray}\int _{B_{1}^{+}}\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D703}}u_{i,j}^{\unicode[STIX]{x1D703}}(f_{t}^{z}hx)\,d\unicode[STIX]{x1D707}(h)\leqslant e^{-(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF})t}\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D703}}u_{i,j}(x)^{\unicode[STIX]{x1D703}}+\unicode[STIX]{x1D716}_{i,j}u(x)+b^{\prime \prime },\end{eqnarray}$$

where $b^{\prime \prime }$ is some constant depending on $t$. Add the above inequalities up over all indices $i,j$, we have

(4.10)$$\begin{eqnarray}\displaystyle \int _{B_{1}^{+}}u(f_{t}^{z}hx)\,dh\leqslant e^{-(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF})t}u(x)+\biggl(\mathop{\sum }_{i,j}\unicode[STIX]{x1D716}_{i,j}\biggr)u(x)+b, & & \displaystyle\end{eqnarray}$$

where $b$ is a constant depending on $t$. In view of (4.10), for all $\unicode[STIX]{x1D716}$ sufficiently small so that $\sum _{i,j}\unicode[STIX]{x1D716}_{i,j}\leqslant e^{-(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF})t}$, one has (4.6) holds.◻

5 Applications of the uniform contraction property

In this section we will introduce some auxiliary sets closely related to $\mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})$ and study them using the uniform contraction property of the EM height function established in Lemma 4.4. To be specific, we will prove some covering results for these auxiliary sets in Proposition 5.1 and these covering results will play an important role in bounding the Hausdorff dimension of $\mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})$.

Let $\unicode[STIX]{x1D6FC}_{1}$ be a contraction rate of the dynamical system $(G/\unicode[STIX]{x1D6E4},F^{+})$ given by (4.4) and let $\unicode[STIX]{x1D706}$ be the top Lyapunov exponent of $A_{F}$ in the representation $(\text{Ad},\mathfrak{g})$. We also fix an auxiliary $\unicode[STIX]{x1D6FF}\in (0,\unicode[STIX]{x1D706})$ which will eventually go to zero. Let $T_{\unicode[STIX]{x1D6FF}}^{\prime }>0$ be given by Lemma 3.2 so that for all $t\geqslant T_{\unicode[STIX]{x1D6FF}}^{\prime }$ we have (3.4) and (3.5) hold. We fix $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{1}$,

(5.1)$$\begin{eqnarray}\displaystyle t>\max \{T_{\unicode[STIX]{x1D6FF}}^{\prime },T(\unicode[STIX]{x1D6FC}),\log 4/\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF},\log 2/\unicode[STIX]{x1D706}\}, & & \displaystyle\end{eqnarray}$$

where $T(\unicode[STIX]{x1D6FC})$ is given by Lemma 4.4 and an EM height function $u:G/\unicode[STIX]{x1D6E4}\rightarrow (0,\infty )$ so that Lemma 4.4 holds. Let $\ell \geqslant 1$ so that (4.7) holds for all $z\in B_{1}^{\text{c}}$ if $u(x)\geqslant \ell$. By Lemma 4.4(ii), there exists $C=C(t)\geqslant 1$ such that

(5.2)$$\begin{eqnarray}\displaystyle C^{-1}u(x)\leqslant u(f_{s}hx)\leqslant Cu(x)\quad \text{for all }0\leqslant s\leqslant t,h\in B_{2}^{G}\text{ and }x\in G/\unicode[STIX]{x1D6E4}. & & \displaystyle\end{eqnarray}$$

Note that the logarithm map from the metric space $(B_{1}^{+},\operatorname{dist})$ to the Lie algebra $\mathfrak{g}^{+}$ is a bi-Lipschitz diffeomorphism to its image with respect to the induced Euclidean structure. Therefore $(B_{1}^{+},\operatorname{dist})$ is Besicovitch, see [Reference MattilaMAT95], namely, for any subset $D$ of $B_{1}^{+}$ and a covering of $D$ by balls centered at $D$, there is a finite sub-covering such that each element of $D$ is covered by at most $E^{\prime }$ times. Therefore, there exists $E\geqslant E^{\prime }$ such that, for all $0<r\leqslant 1$, the set $B_{1/2}^{+}$ can be covered by no more than $Er^{-d}$ open balls of radius $r$, where $d=\dim G^{+}$.

We use $|I|$ to denote the cardinality of a finite set $I$. The following is the main result of this section.

Proposition 5.1. Let $x\in G/\unicode[STIX]{x1D6E4}$. There exists $0<\unicode[STIX]{x1D70E}<1$ and $E_{0}\geqslant 1$ such that, for $z\in B_{1}^{\text{c}}$ and $N\in \mathbb{N}$, the set

(5.3)$$\begin{eqnarray}\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},s)\,\stackrel{\text{def}}{=}\,\{h\in B_{1/2}^{+}:|\{1\leqslant n\leqslant N:u(f_{nt}^{z}hx)\geqslant s\}|\geqslant \unicode[STIX]{x1D70E}N\}\end{eqnarray}$$

with $s=C^{2}\ell$ can be covered by no more than $E_{0}e^{(d\unicode[STIX]{x1D706}-\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}(d+\unicode[STIX]{x1D6FC}))tN}$ open balls of radius $e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})tN}$ in $B_{1}^{+}$.

The rest of this section is devoted to show that Proposition 5.1 holds for

(5.4)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}=\frac{(1-\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D6FC}t+\log C}{\unicode[STIX]{x1D6FC}t+\log C}. & & \displaystyle\end{eqnarray}$$

In the rest of this section we fix $z\in B_{1}^{\text{c}}$ and $N\in \mathbb{N}$. We begin with the following simple observation.

Proof. Let $h_{0}$ be the center of $B$ and $h\in B$. It suffices to show that, for all $1\leqslant n\leqslant N$, if $u(f_{nt}^{z}h_{0}x)\geqslant C^{2}\ell$ then $u(f_{nt}^{z}hx)\geqslant C\ell$. By (3.4) we have

$$\begin{eqnarray}\operatorname{dist}(f_{nt}^{z}h_{0},f_{nt}^{z}h)=\operatorname{dist}(1_{G},f_{nt}^{z}hh_{0}^{-1}f_{-nt}^{z})<1.\end{eqnarray}$$

By (5.2)

$$\begin{eqnarray}u(f_{nt}hx)=u(f_{nt}hh_{0}^{-1}f_{-nt}\cdot f_{nt}h_{0}x)\geqslant C^{-1}u(f_{nt}h_{0}x)\geqslant C^{-1}\cdot C^{2}\ell =C\ell .\Box\end{eqnarray}$$

For a subset $I\subset \{1,\ldots ,N\}$, we let

$$\begin{eqnarray}\mathfrak{D}_{x}(z,I,C\ell )=\{h\in B_{1/2}^{+}:u(f_{nt}^{z}hx)\geqslant C\ell \text{ for all }n\in I\}.\end{eqnarray}$$

Since $\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C\ell )=\bigcup _{|I|\geqslant \unicode[STIX]{x1D70E}N}\mathfrak{D}_{x}(z,I,C\ell )$, one has

(5.5)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C\ell ))\leqslant \mathop{\sum }_{|I|\geqslant \unicode[STIX]{x1D70E}N}\unicode[STIX]{x1D707}(\mathfrak{D}_{x}(z,I,C\ell )). & & \displaystyle\end{eqnarray}$$

The following lemma will play an important role in the proof of Proposition 5.1.

Lemma 5.3. Suppose that $I\subset \{1,\ldots ,N\}$ and $|I|\geqslant \unicode[STIX]{x1D70E}N$. Then

(5.6)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(\mathfrak{D}_{x}(z,I,C\ell ))\leqslant C^{2}u(x)e^{-(1-\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D6FC}tN}. & & \displaystyle\end{eqnarray}$$

We fix $I$ as in the statement of Lemma 5.3. Our strategy is to estimate the measure of $\mathfrak{D}_{x}(z,I,C\ell )$ by relating it to a subset coming from random walks on $G/\unicode[STIX]{x1D6E4}$ with alphabet $f_{t}^{z}B_{1}^{+}$. Let $p=\sup I$ and for $1\leqslant k\leqslant p$ let

$$\begin{eqnarray}Z_{k}=\{(h_{1},\ldots ,h_{k})\in (B_{1}^{+})^{k}:u(f_{t}^{z}h_{n}\cdots f_{t}^{z}h_{1}x)\geqslant \ell ~\forall n\in (I\cap [1,k])\}.\end{eqnarray}$$

Define $\unicode[STIX]{x1D702}:(B_{1}^{+})^{p}\rightarrow G^{+}$ by

(5.7)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}(h_{1},\ldots ,h_{p})=\tilde{h}_{p}\cdots \tilde{h}_{1}\quad \text{where }\tilde{h}_{n}=f_{-(n-1)t}^{z}h_{n}f_{(n-1)t}^{z}. & & \displaystyle\end{eqnarray}$$

By (3.4), we know that $\tilde{h}_{n}\in B_{1/4^{n-1}}^{+}$, and hence by the right invariance of $\operatorname{dist}(\cdot ,\cdot )$, the image of $\unicode[STIX]{x1D702}$ is contained in $B_{2}^{+}$. The following two lemmas are needed in the proof of Lemma 5.3.

Proof. Suppose that $\unicode[STIX]{x1D702}(h_{1},\ldots ,h_{p})=h$ where $h_{i}\in B_{1}^{+}$. In view of (5.7), for $1\leqslant n\leqslant p$ we have

$$\begin{eqnarray}f_{nt}^{z}h=f_{nz}\tilde{h}_{p}\cdots \tilde{h}_{1}\quad \text{and}\quad f_{t}^{z}h_{n}\cdots f_{t}^{z}h_{1}=f_{nt}^{z}\tilde{h}_{n}\cdots \tilde{h}_{1}.\end{eqnarray}$$

Therefore,

$$\begin{eqnarray}f_{nt}^{z}h\cdot (f_{t}^{z}h_{n}\cdots f_{t}^{z}h_{1})^{-1}=\mathop{\prod }_{n<i\leqslant p}f_{nt}^{z}\tilde{h}_{i}(f_{nt}^{z})^{-1}\end{eqnarray}$$

which is contained in the image of $\unicode[STIX]{x1D702}$ with $p$ replaced by $n-p$. Since the image of $\unicode[STIX]{x1D702}$ is always contained in $B_{2}^{+}$ and the metric on $G$ is right invariant, we have

$$\begin{eqnarray}\displaystyle \operatorname{dist}(f_{nt}^{z}h,f_{t}^{z}h_{n}\cdots f_{t}^{z}h_{1})=\operatorname{dist}(f_{nt}^{z}h\cdot (f_{t}^{z}h_{n}\cdots f_{t}^{z}h_{1})^{-1},1_{G})<2. & & \displaystyle \nonumber\end{eqnarray}$$

Therefore, by (5.2) we have for $n\in I$

$$\begin{eqnarray}\displaystyle u(f_{t}^{z}h_{n}\cdots f_{t}^{z}h_{1}x)\geqslant C^{-1}u(f_{nt}^{z}hx)\geqslant \ell . & & \displaystyle \nonumber\end{eqnarray}$$

So $(h_{1},\ldots ,h_{p})\in Z_{p}$ and the proof is complete.◻

Let $\widetilde{\unicode[STIX]{x1D707}}_{n}$ be the Radon measure on $G^{+}$ defined by

(5.8)$$\begin{eqnarray}\displaystyle \int _{G^{+}}\unicode[STIX]{x1D711}(h)\,d\widetilde{\unicode[STIX]{x1D707}}_{n}(h)=\int _{B_{1}^{+}}\unicode[STIX]{x1D711}(f_{-nt}^{z}hf_{nt}^{z})\,dh & & \displaystyle\end{eqnarray}$$

for all $\unicode[STIX]{x1D711}\in C_{c}(G^{+})$. For any positive integer $n$ let $\unicode[STIX]{x1D707}_{n}=\widetilde{\unicode[STIX]{x1D707}}_{n-1}\ast \cdots \ast \widetilde{\unicode[STIX]{x1D707}}_{1}\ast \widetilde{\unicode[STIX]{x1D707}}_{0}$ be the measure on $G^{+}$ defined by the $n$ convolutions. Clearly, $\unicode[STIX]{x1D707}_{n}$ is absolutely continuous with respect to $\unicode[STIX]{x1D707}$, and $\unicode[STIX]{x1D707}_{p}$ is the pushforward of $(\unicode[STIX]{x1D707}|_{B_{1}^{+}})^{\otimes p}$ by the map $\unicode[STIX]{x1D702}$. The following lemma shows that $\unicode[STIX]{x1D707}_{n}$ has density bigger than or equal to $1$ at every $h\in B_{1/2}^{+}$.

Proof. The conclusion is clear if $n=1$. Now we assume $n>1$ and let

$$\begin{eqnarray}\unicode[STIX]{x1D708}=\widetilde{\unicode[STIX]{x1D707}}_{n-1}\ast \widetilde{\unicode[STIX]{x1D707}}_{n-2}\ast \cdots \ast \widetilde{\unicode[STIX]{x1D707}}_{1}.\end{eqnarray}$$

It follows from (3.4) and (5.8) that for $k>0$ the probability measure $\widetilde{\unicode[STIX]{x1D707}}_{k}$ is supported on $B_{1/4^{k}}^{+}$. Since the metric on $G^{+}$ is right invariant, the measure $\unicode[STIX]{x1D708}$ is supported on $B_{1/2}^{+}$. Suppose $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D711}\,d\unicode[STIX]{x1D707}$, then $\unicode[STIX]{x1D707}_{n}=\unicode[STIX]{x1D708}\ast \widetilde{\unicode[STIX]{x1D707}}_{0}=\unicode[STIX]{x1D711}\ast \unicode[STIX]{x1D7D9}_{B_{1}^{+}}\,d\unicode[STIX]{x1D707}$. So for any $h\in B_{1/2}^{+}$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D711}\ast \unicode[STIX]{x1D7D9}_{B_{1}^{+}}(h)=\int _{G^{+}}\unicode[STIX]{x1D711}(h_{1})\unicode[STIX]{x1D7D9}_{B_{1}^{+}}(h_{1}^{-1}h)\,d\unicode[STIX]{x1D707}(h_{1})\geqslant \int _{B_{1/2}^{+}}\unicode[STIX]{x1D711}(h_{1})\,d\unicode[STIX]{x1D707}(h_{1})=1.\Box\end{eqnarray}$$

Now we are ready to prove Lemma 5.3.

Proof of Lemma 5.3.

By Lemmas 5.4 and 5.5,

(5.9)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(\mathfrak{D}_{x}(z,I,C\ell ))\leqslant \unicode[STIX]{x1D707}_{p}(\mathfrak{D}_{x}(z,I,C\ell ))\leqslant \unicode[STIX]{x1D707}^{\otimes p}(Z_{p}). & & \displaystyle\end{eqnarray}$$

So it suffices to estimate $\unicode[STIX]{x1D707}^{\otimes p}(Z_{p})$. For $1\leqslant k\leqslant p$ let

$$\begin{eqnarray}s(k)=\int _{Z_{k}}u(f_{t}^{z}h_{k}\cdots f_{t}^{z}h_{1}x)\,d\unicode[STIX]{x1D707}^{\otimes k}(h_{1},\ldots ,h_{k}).\end{eqnarray}$$

Let

(5.10)$$\begin{eqnarray}\displaystyle s(p+1)=\int _{Z_{p}}\biggl[\int _{B_{1}^{+}}u(f_{t}^{z}h_{p+1}f_{t}^{z}h_{p}\cdots f_{t}^{z}h_{1}x)\,d\unicode[STIX]{x1D707}(h_{p+1})\biggr]d\unicode[STIX]{x1D707}^{\otimes p}(h_{1},\ldots ,h_{p}). & & \displaystyle\end{eqnarray}$$

Then, for every $1<k\leqslant p+1$,

$$\begin{eqnarray}\displaystyle s(k)\leqslant \int _{Z_{k-1}}\biggl[\int _{B_{1}^{+}}u(f_{t}^{z}h_{k}f_{t}^{z}h_{k-1}\cdots f_{t}^{z}h_{1}x)\,d\unicode[STIX]{x1D707}(h_{k})\biggr]\,d\unicode[STIX]{x1D707}^{\otimes (k-1)}(h_{1},\ldots ,h_{k-1}). & & \displaystyle \nonumber\end{eqnarray}$$

If $k-1\in I$, then $s(k)\leqslant e^{-\unicode[STIX]{x1D6FC}t}s(k-1)$ by (4.7). If $k-1\not \in I$, then by (5.2) we have $s(k)\leqslant Cs(k-1)$. We apply this estimate to $k=p+1,p,\ldots ,2$; then we have

$$\begin{eqnarray}\displaystyle s(p+1)\leqslant C^{(N-|I|)}e^{-|I|\unicode[STIX]{x1D6FC}t}\int _{B_{1}^{+}}u(f_{t}hx)\,d\unicode[STIX]{x1D707}(h)\leqslant C^{1+(1-\unicode[STIX]{x1D70E})N}e^{-\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FC}tN}u(x), & & \displaystyle \nonumber\end{eqnarray}$$

where in the second inequality we use the assumption $|I|\geqslant \unicode[STIX]{x1D70E}N$ and (5.2).

According to (5.4), we have $C^{\unicode[STIX]{x1D70E}N}e^{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FC}tN}=C^{N}e^{(1-\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D6FC}tN}$, and hence

(5.11)$$\begin{eqnarray}\displaystyle s(p+1)\leqslant C^{-N}e^{-(1-\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D6FC}N}C^{1+N}u(x)=Ce^{-(1-\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D6FC}tN}u(x). & & \displaystyle\end{eqnarray}$$

On the other hand, in view of (5.10), (5.2) and the fact $p=\sup I$ we have

(5.12)$$\begin{eqnarray}\displaystyle s(p+1)\geqslant C^{-1}s(p)\geqslant C^{-1}\ell \cdot \unicode[STIX]{x1D707}^{\otimes p}(Z_{p}). & & \displaystyle\end{eqnarray}$$

Therefore, (5.6) follows from (5.9), (5.11) and (5.12) and the observation $\ell \geqslant 1$.◻

Proof of Proposition 5.1.

As before we fix $z$ and $N$ as in the statement. Let $\unicode[STIX]{x1D70E}$ be as in (5.4). Since $(B_{1}^{+},\operatorname{dist})$ is Besicovitch, there exists a covering $\mathfrak{U}$ of $\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C^{2}\ell )$ by open balls of radius $e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})tN}$ centered at $\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C^{2}\ell )$ such that each element of $\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C^{2}\ell )$ is covered by at most $E$ elements. By Lemma 5.2, each $B\in \mathfrak{U}$ is contained in $\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C\ell )$. So in view of (3.1)

(5.13)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C\ell ))\geqslant \frac{|\mathfrak{U}|}{E}\unicode[STIX]{x1D707}(B_{e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})tN}}^{+})\geqslant \frac{|\mathfrak{U}|}{C_{0}E}e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})dtN}. & & \displaystyle\end{eqnarray}$$

On the other hand, there are $2^{N}$ subsets $I\subset \{1,\ldots ,N\}$. So by (5.5) and Lemma 5.3, we have

(5.14)$$\begin{eqnarray}\unicode[STIX]{x1D707}(\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C\ell ))\leqslant C^{2}2^{N}e^{-(1-\unicode[STIX]{x1D6FF}/2)\unicode[STIX]{x1D6FC}tN}u(x).\end{eqnarray}$$

Note that $2^{N}<e^{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}tN/2}$ by (5.1). This together with (5.14) implies

(5.15)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C\ell ))\leqslant C^{2}e^{-(1-\unicode[STIX]{x1D6FF})\unicode[STIX]{x1D6FC}tN}u(x). & & \displaystyle\end{eqnarray}$$

By (5.13) and (5.15),

$$\begin{eqnarray}|\mathfrak{U}|\leqslant u(x)C_{0}C^{2}E\cdot e^{(d\unicode[STIX]{x1D706}-\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}(d+\unicode[STIX]{x1D6FC}))tN}.\end{eqnarray}$$

The conclusion now follows by taking $E_{0}=u(x)C_{0}C^{2}E$.◻

6 Upper bound of Hausdorff dimension

In this section, we complete the proof of Theorem 1.4. We will use the same notation as in § 5 prior to Proposition 5.1. For $(z,h)\in B_{1}^{\text{c}}B_{1}^{+},\ell ^{\prime }>0$ and $N\in \mathbb{N}$, let $I_{N,\ell ^{\prime }}(z,h)$ denote the set of $n\in \{1,\ldots ,N\}$ satisfying $u(f_{nt}zhx)\geqslant \ell ^{\prime }$. For $x\in G/\unicode[STIX]{x1D6E4}$, let

(6.1)$$\begin{eqnarray}\mathfrak{D}_{x}^{0}(F^{+},N,\unicode[STIX]{x1D70E},\ell ^{\prime })=\{(z,h)\in B_{1/2}^{\text{c}}B_{1/2}^{+}:|I_{N,\ell ^{\prime }}(z,h)|\geqslant \unicode[STIX]{x1D70E}N\}.\end{eqnarray}$$

Proof. Let $0<\unicode[STIX]{x1D70E}<1$ and $E_{0}\geqslant 1$ so that Proposition 5.1 holds. We will show that the lemma holds for this $\unicode[STIX]{x1D70E}$ and some constant $E_{2}$ which will be specified later.

We fix $N\in \mathbb{N}$. We claim that for any metric ball $W=B_{e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})tN}}^{\text{c}}z\subset B_{1}^{\text{c}}$ where $z\in G^{\text{c}}$, we have

(6.2)$$\begin{eqnarray}(\mathfrak{D}_{x}^{0}(F^{+},N,\unicode[STIX]{x1D70E},C^{4}\ell )\cap (WB_{1}^{+}))\subset (W\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C^{2}\ell )).\end{eqnarray}$$

Let $(z_{1},h_{1})\in WB_{1}^{+}$. Suppose that $1\leqslant n\leqslant N$ and $u(f_{nt}z_{1}h_{1}x)\geqslant C^{4}\ell$. In view of (5.2) we have

(6.3)$$\begin{eqnarray}u(f_{nt}^{z}h_{1}x)=u(z^{-1}f_{nt}zh_{1}x)\geqslant C^{-1}u(f_{nt}zh_{1}x)=C^{-1}u(f_{nt}(zz_{1}^{-1})f_{-nt}\cdot f_{nt}z_{1}h_{1}x).\end{eqnarray}$$

By the right invariance of the metric, $\operatorname{dist}(zz_{1}^{-1},1_{G})=\operatorname{dist}(z_{1}z^{-1},1_{G})\leqslant e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})tN}$. So (3.5) implies $\operatorname{dist}(f_{nt}(zz_{1}^{-1})f_{-nt},1_{G})\leqslant e^{-\unicode[STIX]{x1D706}tN}$. This together with (6.3) and (5.2) imply

$$\begin{eqnarray}u(f_{nt}^{z}h_{1}x)\geqslant C^{-2}u(f_{nt}z_{1}h_{1}x)\geqslant C^{2}\ell .\end{eqnarray}$$

In other words, we have proved that if $n\in I_{N,C^{4}\ell }(z_{1},h_{1})$, then $u(f_{nt}^{z}h_{1}x)\geqslant C^{2}\ell$. Therefore, if $(z_{1},h_{1})$ belongs to the left-hand side of (6.2), then it also belongs to the right-hand side.

Since $(B_{1}^{\text{c}},\text{dist})$ is also Besicovitch, there exists $E_{1}\geqslant 1$ such that, for all $0<r\leqslant 1$, $B_{1/2}^{\text{c}}$ can be covered by no more than $E_{1}r^{-d^{\text{c}}}$ open balls of radius $r$ centered in $B_{1/2}^{\text{c}}$. We fix a cover $\mathfrak{U}^{\text{c}}$ of $B_{1/2}^{\text{c}}$ that consists of open balls of radius $e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})Nt}$ with $|\mathfrak{U}^{\text{c}}|\leqslant E_{1}e^{d^{\text{c}}(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})Nt}$. We assume each element of $\mathfrak{U}^{\text{c}}$ is centered in $B_{1/2}^{\text{c}}$ so that it is contained in $B_{1}^{\text{c}}$ by (5.1). Let $W_{z}\in \mathfrak{U}^{\text{c}}$ be a ball centered at $z\in B_{1}^{\text{c}}$. Proposition 5.1 implies that there exists a covering $\mathfrak{U}_{z}$ of $\mathfrak{D}_{x}(z,N,\unicode[STIX]{x1D70E},C^{2}\ell )$ by open balls of radius $e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})tN}$ such that

$$\begin{eqnarray}\displaystyle |\mathfrak{U}_{z}|\leqslant E_{0}e^{(d\unicode[STIX]{x1D706}-\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}(d+\unicode[STIX]{x1D6FC}))tN}. & & \displaystyle \nonumber\end{eqnarray}$$

In view of claim (6.2), the following class of sets

(6.4)$$\begin{eqnarray}\displaystyle \{W_{z}B:W_{z}\in \mathfrak{U}^{\text{c}},B\in \mathfrak{U}_{z}\} & & \displaystyle\end{eqnarray}$$

forms an open cover of $\mathfrak{D}_{x}^{0}(F^{+},N,\unicode[STIX]{x1D70E},C^{4}\ell )$. It is easily checked that there exists $E_{1}^{\prime }\geqslant 1$ not depending on $N$ such that each element $W_{z}B$ of (6.4) can be covered by $E_{1}^{\prime }$ open balls of radius $e^{-(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D6FF})Nt}$ in $G^{\text{c}}G^{+}$. Therefore the lemma holds with $E_{2}=E_{0}E_{1}E_{1}^{\prime }$.◻

Theorem 6.2. For any $x\in G/\unicode[STIX]{x1D6E4}$, the Hausdorff dimension of

$$\begin{eqnarray}\mathfrak{D}_{x}^{0}\,\stackrel{\text{def}}{=}\,\{(z,h)\in B_{1/2}^{\text{c}}B_{1/2}^{+}:zhx\in \mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})\}\end{eqnarray}$$

is at most $d^{\text{c}}+d-\unicode[STIX]{x1D6FC}_{1}/\unicode[STIX]{x1D706}$.

Proof. For each $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{1}$ and $0<\unicode[STIX]{x1D6FF}<1$ we first choose $t>0$, a height function $u$ and $\ell ,C\geqslant 1$ so that Lemma 4.4, (3.4), (3.5), (5.2) and (5.1) hold. Then there exists $0<\unicode[STIX]{x1D70E}<1$ and $E_{2}\geqslant 1$ so that Lemma 6.1 holds.

It follows from parts (i) and (ii) of Lemma 4.4 and the definition of $\mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})$ that

$$\begin{eqnarray}\mathfrak{D}_{x}^{0}\subset \mathop{\bigcup }_{M\geqslant 1}Q_{M}\quad \text{where }Q_{M}=\mathop{\bigcap }_{N\geqslant M}\mathfrak{D}_{x}^{0}(F^{+},N,\unicode[STIX]{x1D70E},C^{4}\ell ).\end{eqnarray}$$

Recall that for any metric space $S$, the Hausdorff dimension of $S$ is less than or equal to the lower box dimension, see e.g. [Reference Bishop and PeresBP17, (1.2.3)]. So Lemma 6.1 implies

$$\begin{eqnarray}\displaystyle \dim _{H}Q_{M} & {\leqslant} & \displaystyle \liminf _{N\rightarrow \infty }\frac{[d^{\text{c}}\unicode[STIX]{x1D706}+d\unicode[STIX]{x1D706}-\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}(d+d^{\text{c}}+\unicode[STIX]{x1D6FC})]tN+\log E_{2}}{\unicode[STIX]{x1D706}tN}\nonumber\\ \displaystyle & = & \displaystyle d^{\text{c}}+d-\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D706}}+\unicode[STIX]{x1D6FF}\frac{d+d^{\text{c}}+\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D706}}.\nonumber\end{eqnarray}$$

Therefore

$$\begin{eqnarray}\dim _{H}\mathfrak{D}_{x}^{0}\leqslant d^{\text{c}}+d-\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D706}}+\unicode[STIX]{x1D6FF}\frac{d+d^{\text{c}}+\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

The conclusion follows by first letting $\unicode[STIX]{x1D6FF}\rightarrow 0$ and then letting $\unicode[STIX]{x1D6FC}\rightarrow \unicode[STIX]{x1D6FC}_{1}$.◻

Proof. Note that, by Lemma 3.1,

$$\begin{eqnarray}\operatorname{dist}(f_{t}hx,f_{t}x)\leqslant \operatorname{dist}(f_{t}hf_{-t},1_{G})\rightarrow 0\end{eqnarray}$$

as $t\rightarrow \infty$. Therefore the lemma holds.◻

Proof of Theorem 1.4.

We will show that

$$\begin{eqnarray}\dim _{H}\mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})\leqslant d^{-}+d^{\text{c}}+d-\frac{\unicode[STIX]{x1D6FC}_{1}}{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

In view of the local nature of the Hausdorff dimension and the definition of the metric on $G/\unicode[STIX]{x1D6E4}$, it suffices to prove that for any $x\in G/\unicode[STIX]{x1D6E4}$

$$\begin{eqnarray}\dim _{H}\{g\in B_{r}^{G}:gx\in \mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})\}\leqslant d^{-}+d^{\text{c}}+d-\frac{\unicode[STIX]{x1D6FC}_{1}}{\unicode[STIX]{x1D706}},\end{eqnarray}$$

where $r<1$ so that $B_{r}^{G}\subset B_{1}^{-}B_{1/2}^{\text{c}}B_{1/2}^{+}$. Recall that we assume the logarithm map on $B_{1}^{G}$ is bi-Lipschitz onto its image in $\mathfrak{g}$ with respect to the metric given by the Euclidean structure. So Marstrand’s product theorem [Reference Bishop and PeresBP17, Theorem 3.2.1] implies

$$\begin{eqnarray}\dim B_{1}^{-}S\leqslant \dim S+d^{-}\end{eqnarray}$$

for any $S\subset B_{1/2}^{\text{c}}B_{1/2}^{+}$. In particular, Lemma 6.3 implies

$$\begin{eqnarray}\{g\in B_{r}^{G}:gx\in \mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})\}\subset B_{1}^{-}\mathfrak{D}_{x}^{0}\end{eqnarray}$$

whose Hausdorff dimension is bounded from above by $\dim _{H}\mathfrak{D}_{x}^{0}+d^{-}$. In view of Theorem 6.2, the Hausdorff dimension of $\mathfrak{D}(F^{+},G/\unicode[STIX]{x1D6E4})$ is at most $d+d^{\text{c}}+d^{-}-\unicode[STIX]{x1D6FC}_{1}/\unicode[STIX]{x1D706}$, which is strictly less than the manifold dimension of $G/\unicode[STIX]{x1D6E4}$.◻