On denseness of horospheres in higher rank homogeneous spaces

Abstract

Let $ G $ be a connected semisimple real algebraic group and $\Gamma <G$ be a Zariski dense discrete subgroup. Let N denote a maximal horospherical subgroup of G, and $P=MAN$ the minimal parabolic subgroup which is the normalizer of N. Let $\mathcal E$ denote the unique P-minimal subset of $\Gamma \backslash G$ and let $\mathcal E_0$ be a $P^\circ $-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary $ G/P $ and show that the following are equivalent for any $[g]\in \mathcal E_0$:

1. (1) $gP\in G/P$ is a horospherical limit point;

2. (2) $[g]NM$ is dense in $\mathcal E$;

3. (3) $[g]N$ is dense in $\mathcal E_0$.

The equivalence of items (1) and (2) is due to Dal’bo in the rank one case. We also show that unlike convex cocompact groups of rank one Lie groups, the $NM$-minimality of $\mathcal E$ does not hold in a general Anosov homogeneous space.

1 Introduction

Let G be a connected semisimple real algebraic group. Let $(X, d)$ denote the associated Riemannian symmetric space. Let $P=MAN$ be a minimal parabolic subgroup of G with fixed Langlands decomposition, where A is a maximal real split torus of G, M the maximal compact subgroup of P commuting with A, and N the unipotent radical of P. Note that N is a maximal horospherical subgroup of G, which is unique up to conjugations.

Fix a positive Weyl chamber $\mathfrak a^+\subset \log A$ so that $\log N$ consists of positive root subspaces, and we set $A^+=\exp \mathfrak a^+$ . This means that N is a contracting horospherical subgroup in the sense that for any a in the interior of $A^+$ ,

$$ \begin{align*}N =\{g\in G: a^{-n} g a^n\to e\text{ as}\ n\to +\infty\}.\end{align*} $$

Let $\Gamma $ be a Zariski dense discrete subgroup of G. In this paper, we are interested in the topological behavior of the action of the horospherical subgroup N on $\Gamma \backslash G$ via the right translations. When $\Gamma <G$ is a cocompact lattice, every N-orbit is dense in $\Gamma \backslash G$ , that is, the N-action on $\Gamma \backslash G$ is minimal. This is due to Hedlund [11] for $G=\operatorname {PSL}_2(\mathbb R)$ and to Veech [19] in general. Dani gave a full classification of possible orbit closures of N-action for any lattice $\Gamma <G$ [8].

For a general discrete subgroup $\Gamma <G$ , the quotient space $\Gamma \backslash G$ does not necessarily admit a dense N-orbit, even a dense $NM$ -orbit, for instance in the case where $\Gamma $ does not have a full limit set. Let $\mathcal F$ denote the Furstenberg boundary $G/P$ . We denote by $\Lambda =\Lambda _\Gamma $ the limit set of $\Gamma $ ,

$$ \begin{align*}\Lambda=\Big\{\lim_{i\to \infty} \gamma_i(o) \in \mathcal F : \gamma_i\in \Gamma\Big\},\end{align*} $$

where $o\in X$ and the convergence is understood as in Definition 2.2. This definition is independent of the choice of $o\in X$ . The limit set $\Lambda $ is known to be the unique $\Gamma $ -minimal subset of $\mathcal F$ (see [1, 9, 15]). Thus, the set

$$ \begin{align*}\mathcal E=\{[g]\in \Gamma\backslash G: gP\in \Lambda\}\end{align*} $$

is the unique P-minimal subset of $\Gamma \backslash G$ . For a given point $[g]\in \mathcal E$ , the topological behavior of the horospherical orbit $[g]N$ (or of $[g]NM$ ) is closely related to the ways in which the orbit $\Gamma (o)$ approaches $gP$ along its limit cone. The limit cone $\mathcal L=\mathcal L_\Gamma $ of $\Gamma $ is defined as the smallest closed cone of $\mathfrak a^+$ containing the Jordan projection $\unicode{x3bb} (\Gamma )$ . It is a convex cone with non-empty interior: $\operatorname {int} \mathcal L\ne \emptyset $ [1]. If $\operatorname {rank } G=1$ , then $\mathcal L=\mathfrak a^+$ . In higher ranks, the limit cone of $\Gamma $ depends more subtly on $\Gamma $ .

1.1 Horospherical limit points

Recall that in the rank one case, a horoball in X based at $\xi \in \mathcal F$ is a subset of the form $gN (\exp \mathfrak a^+ )( o)$ , where $g\in G$ is such that $\xi =gP$ [5]. Our generalization to higher rank of the notion of a horospherical limit point involves the limit cone of $\Gamma $ . By a $\Gamma $ -tight horoball based at $\xi \in \mathcal F$ , we mean a subset of the form $\mathcal H_{\xi }=g N (\exp \mathcal C) (o)$ , where $g\in G$ is such that $\xi =gP$ and $\mathcal C$ is a closed cone contained in $\operatorname {int} \mathcal L\cup \{0\}$ . For $T>0$ , we write

$$ \begin{align*}\mathcal H_{\xi}(T)= gN (\exp (\mathcal C-\mathcal C_T))o,\end{align*} $$

where $\mathcal C_T=\{u\in \mathcal C: \|u\|<T\}$ for a Euclidean norm $\|\cdot \|$ on $\mathfrak a$ .

Definition 1.1. We call a limit point $\xi \in \Lambda $ a horospherical limit point of $\Gamma $ if one of the following equivalent conditions holds:

• • there exists a $\Gamma $ -tight horoball $\mathcal H_\xi $ based at $\xi $ such that for any $T>1$ , $\mathcal H_\xi (T)$ contains some point of $\Gamma (o)$ ;

• • there exist a closed cone $\mathcal C \subset {\operatorname {int} \mathcal L} \cup \{0\}$ and a sequence $ \gamma _j \in \Gamma $ satisfying that $\beta _\xi (o,\gamma _jo) \in \mathcal C$ for all $j\ge 1$ and $ \beta _\xi (o,\gamma _jo) \to \infty $ as $j\to \infty $ , where $\beta $ denotes the $\mathfrak a$ -valued Busemann map (Definition 2.3).

See Lemma 3.3 for the equivalence of the above two conditions. We denote by

$$ \begin{align*}\Lambda_h\subset \Lambda\end{align*} $$

the set of all horospherical limit points of $\Gamma $ . The attracting fixed point $y_\gamma $ of a loxodromic element $\gamma \in \Gamma $ whose Jordan projection $\unicode{x3bb} (\gamma )$ belongs to $\operatorname {int} \mathcal L$ is always a horospherical limit point (Lemma 3.5). Moreover, for any $u\in \operatorname {int} \mathcal L$ , any u-directional radial limit point $\xi $ (i.e. $\xi =gP$ for some $g\in G$ such that $\limsup _{t\to \infty } \Gamma g \exp (tu)\ne \emptyset $ ) is also a horospherical limit point (Lemma 5.3).

Remarks 1.2

1. (1) There exists a notion of horospherical limit points in the geometric boundary associated to a symmetric space, see [10]. When $\operatorname {rank }G \geq 2 $ , this notion and the one considered here are different.

2. (2) Unlike the rank one case, a sequence $\gamma _i(o)\in \mathcal H_{\xi }(T_i)$ , with $T_i\to \infty $ , does not necessarily converge to $\xi $ for a $\Gamma $ -tight horoball $\mathcal H_\xi $ based at $\xi $ . It is hence plausible that a general discrete group $ \Gamma $ would support a horospherical limit point outside of its limit set.

1.2 Denseness of horospheres

The following theorem generalizes Dal’bo’s theorem [5] to discrete subgroups in higher rank semisimple Lie groups.

Theorem 1.3. Let $\Gamma <G$ be a Zariski dense discrete subgroup. For any $[g]\in \mathcal E$ , the following are equivalent:

1. (1) $gP\in \Lambda _h$ ;

2. (2) $[g] NM$ is dense in $\mathcal E$ .

Remarks 1.4. Conze and Guivarc’h considered the notion of a horospherical limit point for Zariski dense discrete subgroups $\Gamma $ of $\operatorname {SL}_d(\mathbb R)$ using the description of $\operatorname {SL}_d(\mathbb R)/P$ as the full flag variety and the standard linear action of $\Gamma $ on $\mathbb R^d$ [4]. By duality, this notion coincides with ours and hence the special case of Theorem 1.3 for $G=\operatorname {SL}_d(\mathbb R)$ also follows from [4, Theorem 4.2]. (However the claim in [4, Theorem 6.3] is incorrect.)

To extend Theorem 1.3 to N-orbits, we fix a $P^\circ $ -minimal subset $\mathcal E_0$ of $\Gamma \backslash G$ , where $P^\circ $ denotes the identity component of P. Clearly, $\mathcal E_0\subset \mathcal E$ . Since $P=P^\circ M$ , any $P^\circ $ -minimal subset is a translate of $\mathcal E_0$ by an element of the finite group $M^\circ \backslash M$ , where $ M^\circ $ is the identity component of $ M $ . Denote by $ \mathfrak D_\Gamma = \{\mathcal E_0,\ldots , \mathcal E_p\} $ the finite collection of all $ P^\circ $ -minimal sets in $ \mathcal E $ . To understand N-orbit closures, it is hence sufficient to restrict to $\mathcal E_0$ .

The following is a refinement of Theorem 1.3.

Theorem 1.5. Let $\Gamma <G$ be a Zariski dense discrete subgroup. For any $[g]\in \mathcal E_0$ , the following are equivalent:

1. (1) $gP\in \Lambda _h$ ;

2. (2) $[g] N$ is dense in $\mathcal E_0$ .

Remark 1.6. We may consider horospherical limit points outside the context of $\Lambda $ . In this case, our proofs of Theorems 1.3 and 1.5 show that if $gP\in \mathcal F$ is a horospherical limit point, then the closures of $[g]MN$ and $[g]N$ contain $\mathcal E$ and $\mathcal E_i $ for some $ \mathcal E_i \in \mathfrak D_\Gamma $ , respectively.

For $G=\operatorname {SO}^\circ ({n,1})$ , $n\ge 2$ , Theorem 1.5 was proved in [16]. When G has rank one and $\Gamma <G$ is convex cocompact, every limit point is horospherical and Winter’s mixing theorem [20] implies the N-minimality of $\mathcal E_0$ .

1.3 Directional horospherical limit points

We also consider the following seemingly much stronger notion.

Definition 1.7. For $u\in \mathfrak a^+$ , a point $ \xi \in \mathcal F$ is called $ u $ -horospherical if there exists a sequence $ \gamma _j \in \Gamma $ such that $\sup _j \| \beta _\xi (o,\gamma _jo)- \mathbb R_+ u\|<\infty $ and $ \beta _\xi (o,\gamma _jo)\to \infty $ as $j\to ~\infty $ .

Denote by $ \Lambda _h(u) $ the set of $ u $ -horospherical limit points. Surprisingly, it turns out that every horospherical limit point is u-horospherical for all $u\in \operatorname {int}\mathcal L$ .

Theorem 1.8. For all $u\in \operatorname {int} \mathcal L$ , we have

$$ \begin{align*}\Lambda_h=\Lambda_h(u).\end{align*} $$

1.4 Existence of non-dense horospheres

A finitely generated subgroup $\Gamma <G$ is called an Anosov subgroup (with respect to P) if there exists $C>0$ such that for all $\gamma \in \Gamma $ , $\alpha (\mu (\gamma ))\ge C|\gamma | -C$ for all simple roots $\alpha $ of $(\mathfrak g, \mathfrak a^+)$ , where $\mu (\gamma )\in \mathfrak a^+$ denotes the Cartan projection of $\gamma $ and $|\gamma |$ is the word length of $\gamma $ with respect to a fixed finite generating set of $\Gamma $ .

For Zariski dense Anosov subgroups of G, almost all $NM$ -orbits are dense in $\mathcal E$ and almost all N-orbits are dense in $\mathcal E_0$ with respect to any Patterson–Sullivan measure on $\Lambda $ [15, 14]. In particular, the set of all horospherical limit points has full Patterson–Sullivan measures.

However, as Anosov subgroups are regarded as higher rank generalizations of convex cocompact subgroups, it is a natural question whether the minimality of the $NM$ -action persists in the higher rank setting. It turns out that it is not the case. Our example is based on Thurston’s theorem [18, Theorem 10.7] together with the following observation on the implication of the existence of a Jordan projection of an element of $\Gamma $ lying in the boundary $\partial \mathcal L$ of the limit cone.

Proposition 1.9. Let $\Gamma <G$ be a Zariski dense discrete subgroup. For any loxodromic element $\gamma \in \Gamma $ , we have

$$ \begin{align*} \unicode{x3bb}(\gamma)\in \operatorname{int} \mathcal L \quad \text{if and only if} \; \{y_{\gamma}, y_{\gamma^{-1}}\} \subset \Lambda_h ,\end{align*} $$

where $y_\gamma $ and $y_{\gamma ^{-1}}$ denote the attracting fixed points of $\gamma $ and $\gamma ^{-1}$ , respectively.

In particular, if $\unicode{x3bb} (\Gamma )\cap \partial \mathcal L\ne \emptyset $ , then $\Lambda \ne \Lambda _h$ and hence there exists a non-dense $NM$ -orbit in $\mathcal E$ .

Thurston’s work [18] provides many examples of Anosov subgroups satisfying that $\unicode{x3bb} (\Gamma )\cap \partial \mathcal L\ne \emptyset $ . To describe them, let $\Sigma $ be a a torsion-free cocompact lattice of $\operatorname {PSL}_2(\mathbb R)$ and let $\pi \kern1.3pt{:}\kern1.3pt\Sigma \kern1.3pt{\to}\kern1.3pt \operatorname {PSL}_2(\mathbb R)$ be a discrete faithful representation. Let ${0\kern1.3pt{<}\kern1.3pt d_-(\pi ) \kern1.3pt{\le}\kern1.3pt d_+ (\pi ) \kern1.3pt{<}\kern1.3pt\infty }$ be the minimal and maximal geodesic stretching constants:

(1.1) $$ \begin{align} d_+(\pi) =\sup_{\sigma\in \Sigma-\{e\} }\frac {\ell(\pi(\sigma))}{\ell(\sigma)} \quad \text{and} \quad d_-(\pi) =\inf_{\sigma\in \Sigma-\{e\}}\frac{\ell(\pi(\sigma))}{\ell(\sigma)}, \end{align} $$

where $\ell (\sigma )$ denotes the length of the closed geodesic in the hyperbolic manifold $\Sigma \backslash \mathbb H^2$ corresponding to $\sigma $ and $\ell (\pi (\sigma ))$ is defined similarly.

Consider the following self-joining subgroup:

$$ \begin{align*}\Gamma_\pi:=(\text{id}\times \pi)(\Sigma)=\{(\sigma, \pi(\sigma)):\sigma\in \Sigma\} <\operatorname{PSL}_2(\mathbb R) \times \operatorname{PSL}_2(\mathbb R).\end{align*} $$

It is easy to see that $\Gamma $ is an Anosov subgroup of $G=\operatorname {PSL}_2(\mathbb R) \times \operatorname {PSL}_2(\mathbb R)$ . Moreover, when $\pi $ is not a conjugate by a Möbius tranformation, $\Gamma _\pi $ is Zariski dense in G (cf. [12, Lemma 4.1]). Identifying $\mathfrak a=\mathbb R^2$ , the Jordan projection $\unicode{x3bb} (\gamma _\pi )$ of $\gamma _\pi =(\sigma , \pi (\sigma ))\in \Gamma _\pi $ is given by $(\ell (\sigma ), \ell (\pi (\sigma )))\in \mathbb R^2$ . Hence, the limit cone $\mathcal L$ of $\Gamma _\pi $ is given by

$$ \begin{align*}\mathcal L:=\{(v_1,v_2)\in \mathbb R_{\ge 0}^2: d_-(\pi) v_1\le v_2\le d_+(\pi) v_1\}. \end{align*} $$

Thurston [18, Theorem 10.7] showed that $d_+(\pi )$ is realized by a simple closed geodesic of $\Sigma \backslash \mathbb H^2$ in most of the cases, which hence provides infinitely many examples of $\Gamma _\pi $ which satisfy $\unicode{x3bb} (\Gamma _\pi )\cap \partial \mathcal L\ne \emptyset $ . Therefore, Proposition 1.9 implies (in this case, we have $NM=N)$ the following corollary.

Corollary 1.10. There are infinitely many non-conjuagte Zariski dense Anosov subgroups $\Gamma _{\pi }<\operatorname {PSL}_2(\mathbb R) \times \operatorname {PSL}_2(\mathbb R)$ with non-dense $NM$ -orbits in $\mathcal E$ .

We close the introduction by the following question (cf. [13, 17]).

Question 1.11. For a simple real algebraic group G with $\operatorname {rank} G\ge 2$ , is every discrete subgroup $\Gamma <G$ with $\Lambda =\Lambda _h=\mathcal F$ necessarily a cocompact lattice in G?

2 Preliminaries

Let G be a connected, semisimple real algebraic group. We fix, once and for all, a Cartan involution $\theta $ of the Lie algebra $\mathfrak {g}$ of G, and decompose $\mathfrak {g}$ as $\mathfrak g=\mathfrak k\oplus \mathfrak {p}$ , where $\mathfrak {k}$ and $\mathfrak {p}$ are the $+ 1$ and $-1$ eigenspaces of $\theta $ , respectively. We denote by K the maximal compact subgroup of G with Lie algebra $\mathfrak {k}$ .

Choose a maximal abelian subalgebra $\mathfrak a$ of $\mathfrak p$ . Choosing a closed positive Weyl chamber $\mathfrak a^+$ of $\mathfrak a$ , let $A:=\exp \mathfrak a$ and $A^+=\exp \mathfrak a^+$ . The centralizer of A in K is denoted by M, and we set N to be the maximal contracting horospherical subgroup: for $a\in \operatorname {int} A^+$ ,

$$ \begin{align*}N =\{g\in G: a^{-n} g a^n\to e\text{ as}\ n\to +\infty\}.\end{align*} $$

We set $P=MAN$ , which is the unique minimal parabolic subgroup of G, up to conjugation.

For $u\in \mathfrak a$ , we write $a_u=\exp u\in A$ . We denote by $\|\cdot \|$ the norm on $\mathfrak g$ induced by the Killing form. Consider the Riemannian symmetric space $X:=G/K$ with the metric induced from the norm $\|\cdot \|$ on $\mathfrak g$ and $o=K\in X$ .

Let $\mathcal {F}=G/P$ denote the Furstenberg boundary. Since K acts transitively on $ \mathcal {F} $ and $K\cap P=M$ , we may identify $\mathcal {F}=K/M$ . We denote by $\mathcal F^{(2)}$ the unique open G-orbit in $\mathcal {F} \times \mathcal {F}$ .

Denote by $w_0\in K$ the unique element in the Weyl group such that $\operatorname {Ad}_{w_0}\mathfrak a^+= -\mathfrak a^+$ ; it is the longest Weyl element. We then have $\check {P}:=w_0 P w_0^{-1}$ is an opposite parabolic subgroup of G, with $ \check {N} $ its unipotent radical. The map $\operatorname {i}=-\operatorname {Ad}_{w_0}: \mathfrak a^+\to \mathfrak a^+$ is called the opposition involution.

For $g\in G$ , we consider the following visual maps:

$$ \begin{align*}g^+:=gP\in \mathcal F\quad\text{and}\quad g^-:=gw_0P\in \mathcal F.\end{align*} $$

Then $\mathcal F^{(2)}=\{(g^+, g^-)\in \mathcal F\times \mathcal F: g\in G\}$ .

Any element $ g \in G $ can be uniquely decomposed as the commuting product $ g_h,g_e,g_u $ , where $ g_h $ , $ g_e $ , and $ g_u $ are hyperbolic, elliptic, and unipotent elements, respectively. The Jordan projection of g is defined as the element $\unicode{x3bb} (g) \in \mathfrak a^+ $ satisfying $ g_h= \varphi \exp \unicode{x3bb} (g) \varphi ^{-1} $ for some $\varphi \in G$ .

An element $g\in G$ is called loxodromic if $ \unicode{x3bb} (g) \in \operatorname {int} \mathfrak a^+ $ ; in this case, $g_u$ is necessarily trivial. For a loxodromic element $g\in G$ , the point $\varphi ^+ \in \mathcal {F}$ is called the attracting fixed point of g, which we denote by $y_g$ . For any loxodromic element $g\in G$ and $\xi \in \mathcal F$ with $(\xi , y_{g^{-1}})\in \mathcal F^{(2)}$ , we have $\lim _{k\to \infty } g^k \xi = y_{g}$ and the convergence is uniform on compact subsets.

Note that for any loxodromic element $g\in G$ ,

$$ \begin{align*}\unicode{x3bb}(g^{-1})= \operatorname{i}\unicode{x3bb}(g).\end{align*} $$

Let $\Gamma <G$ be a Zariski dense discrete subgroup of G. The limit cone $\mathcal L=\mathcal L_\Gamma $ of $\Gamma $ is the smallest closed cone of $\mathfrak a^+$ containing $\unicode{x3bb} (\Gamma )$ . It is a convex cone with non-empty interior [1].

We will use the following simple lemma.

Lemma 2.1. For any $ v \in \unicode{x3bb} (\Gamma ) $ and $ \zeta \in \mathcal {F} $ , there exists a loxodromic element $ \gamma \in \Gamma $ with $ \unicode{x3bb} (\gamma )=v $ and a neighborhood $ U $ of $ \zeta $ in $ \mathcal {F} $ such that $ \{y_\gamma \}\times U $ is a relatively compact subset of $\mathcal {F}^{(2)} $ and as $k\to \infty $ ,

$$ \begin{align*} \gamma^{-k} \zeta \to y_{\gamma^{-1}} \quad \text{uniformly on}\ U. \end{align*} $$

Proof. Let $\zeta \in \mathcal F$ . Choose $\gamma _1\in \Gamma $ such that $\unicode{x3bb} (\gamma _1)=v$ . Since the set of all loxodromic elements of $\Gamma $ is Zariski dense in G [2] and $\mathcal F^{(2)}$ is Zariski open in $\mathcal F\times \mathcal F$ , there exists $\gamma _2\in \Gamma $ such that $(\zeta , \gamma _2 y_{\gamma _1})\in \mathcal F^{(2)}$ . Let $\gamma =\gamma _2 \gamma _1\gamma _2^{-1}$ , so that $y_{\gamma }=\gamma _2y_{\gamma _1}$ . It now suffices to take any neighborhood U of $\zeta $ such that $U\times \{ \gamma _2y_{\gamma _1}\}$ is a relatively compact subset of $\mathcal F^{(2)}$ .

2.1 Convergence of a sequence in X to $\mathcal F$

By the Cartan decomposition $G=KA^+K$ , for $g\in G$ , we may write

$$ \begin{align*}g=\kappa_1(g)\exp (\mu(g))\kappa_2(g)\in KA^+K,\end{align*} $$

where $\mu (g)\in \mathfrak a^+$ , called the Cartan projection of g, is uniquely determined, and $\kappa _1(g), \kappa _2(g)\in K$ . If $\mu (g)\in \operatorname {int} \mathfrak a^+$ , then $[\kappa _1(g)]\in K/M=\mathcal F$ is uniquely determined.

Let $\Pi $ be the set of simple roots for $(\mathfrak g, \mathfrak a)$ . For a sequence $g_i\to G$ , we say $g_i\to \infty $ regularly if $\alpha (\mu (g_i))\to \infty $ for all $\alpha \in \Pi $ . Note that if $g_i\to \infty $ regularly, then for all sufficiently large i, $\mu (g_i)\in \operatorname {int}\mathfrak a^+$ and hence $[\kappa _1(g_i)]$ is well defined.

Definition 2.2. A sequence $p_i \in X$ is said to converge to $\xi \in \mathcal F$ if there exists $g_i\to \infty $ regularly in G with $p_i=g_i(o)$ and $\lim _{i\to \infty }[\kappa _1(g_i)]=\xi $ .

2.2 $P^\circ $ -minimal subsets

We denote by $\Lambda \subset \mathcal F$ the limit set of $\Gamma $ , which is defined as

(2.1) $$ \begin{align} \Lambda=\{\lim \gamma_i(o): \gamma_i\in \Gamma\}.\end{align} $$

For a non-Zariski dense subgroup, $\Lambda $ may be an empty set. For $\Gamma <G$ Zariski dense, this is the unique $\Gamma $ -minimal subset of $\mathcal F$ [1, 15].

It follows that the following set $\mathcal E$ is the unique P-minimal subset of $\Gamma \backslash G$ :

$$ \begin{align*}\mathcal E=\{[g]\in \Gamma\backslash G: g^+\in \Lambda\}.\end{align*} $$

Let $P^\circ $ denote the identity component of P. Then $\mathcal E$ is a disjoint union of at most $[P: P^{\circ }]$ -number of $P^\circ $ -minimal subsets. We fix one $P^{\circ }$ -minimal subset $\mathcal E_0$ once and for all. Note that any $P^{\circ }$ -minimal subset is then of the form $\mathcal E_0 m$ for some $m\in M$ . We set

(2.2) $$ \begin{align} \Omega:=\{[g]\in \Gamma\backslash G: g^+, g^-\in \Lambda\}\quad\text{and}\quad \Omega_0:=\Omega\cap \mathcal E_0 .\end{align} $$

2.3 Busemann map

The Iwasawa cocycle $\sigma : G\times \mathcal {F} \to \mathfrak a$ is defined as follows: for $(g, \xi )\in G\times \mathcal {F} $ with $\xi =[k]$ for $k\in K$ , $\exp \sigma (g,\xi )$ is the A-component of $g k$ in the $KAN$ decomposition, that is,

$$ \begin{align*}gk\in K \exp (\sigma(g, \xi)) N.\end{align*} $$

The $\mathfrak a$ -valued Busemann function $\beta : \mathcal {F} \times X \times X \to \mathfrak a $ is defined as follows: for $\xi \in \mathcal {F} $ and $g, h\in G$ ,

$$ \begin{align*}\beta_\xi ( ho, go):=\sigma (h^{-1}, \xi)-\sigma(g^{-1}, \xi).\end{align*} $$

We note that for any $g\in G$ , $\xi \in \mathcal F$ , and $x,y, z\in X$ ,

(2.3) $$ \begin{align} \beta_\xi(x,y)=\beta_{g\xi}(gx, gy)\quad\text{and}\quad \beta_\xi(x, y)=\beta_\xi(x,z)+\beta_\xi(z, y).\end{align} $$

In particular, $ \beta _\xi (o,go) \in \mathfrak a $ is defined by

(2.4) $$ \begin{align} g^{-1}k_\xi \in K \exp(-\beta_\xi(o, go)) N, \end{align} $$

and hence $\beta _P (o, a_u o)=u$ for any $u\in \mathfrak a$ . For $h, g\in G$ , we set $\beta _\xi (h, g):=\beta _{\xi }(ho, go)$ .

2.4 Shadows

For $ q\in X$ and $r>0$ , we set $B(q,r)=\{x\in X: d(x, q)\le r\}$ . For $p=g(o)\in X$ , the shadow of the ball $B(q,r)$ viewed from p is defined as

$$ \begin{align*}O_r(p,q):=\{(gk)^+\in \mathcal F: k\in K,\; gk\operatorname{int} A^+o\cap B(q,r)\ne \emptyset\}.\end{align*} $$

Similarly, for $\xi \in \mathcal F$ , the shadow of the ball $B(q,r)$ as viewed from $\xi $ is

$$ \begin{align*}O_r(\xi,q):=\{h^+\in \mathcal F: h\in G\text{ satisfies } h^-=\xi,\, ho\in B(q,r) \}. \end{align*} $$

Lemma 2.3. [15, Lemmas 5.6 and 5.7]

1. (1) There exists $\kappa>0$ such that for any $g\in G$ and $r>0$ ,

   $$ \begin{align*}\sup_{\xi \in O_r(g(o),o)}\|\beta_{\xi}(g(o),o)-\mu(g^{-1})\|\le \kappa r .\end{align*} $$

2. (2) If a sequence $p_i\in X$ converges to $\xi \in \mathcal F$ , then for any $0<\varepsilon <r$ , we have

   $$ \begin{align*}O_{r-\varepsilon} (p_i, o)\subset O_r(\xi, o)\subset O_{r+\varepsilon} (p_i, o)\end{align*} $$
   for all sufficiently large i.

3 Horospherical limit points

Let $\Gamma <G$ be a Zariski dense discrete subgroup. A $\Gamma $ -tight horoball based at $\xi \in \mathcal F$ is a subset of the form $\mathcal H_{\xi }=g N (\exp \mathcal C) (o)$ , where $g\in G$ is such that $\xi =gP$ and $\mathcal C$ is a closed cone contained in $\operatorname {int} \mathcal L\cup \{0\}$ . For $T>0$ , we write $\mathcal H_{\xi }(T)= gN (\exp (\mathcal C-\mathcal C_T))o$ . We recall the definition from the introduction.

Definition 3.1. We say that $\xi \in \mathcal F$ is a horospherical limit point of $\Gamma $ if there exists a $\Gamma $ -tight horoball $\mathcal H_\xi $ based at $\xi $ such that $\mathcal H_\xi (T)\cap \Gamma (o)\ne \emptyset $ for all $T>1$ .

In this section, we provide a mostly self-contained proof of the following theorem.

Theorem 3.2. Let $[g]\in \mathcal E$ . The following are equivalent:

1. (1) $g^+=gP\in \Lambda $ is a horospherical limit point;

2. (2) $[g]NM$ is dense in $\mathcal E$ .

The main external ingredient in our proof is the density of the group generated by the Jordan projection $ \unicode{x3bb} (\Gamma )$ , due to Benoist [2], that is,

$$ \begin{align*} \mathfrak a = \overline{\langle \unicode{x3bb}(\Gamma) \rangle} \end{align*} $$

for every Zariski dense discrete subgroup $ \Gamma < G $ . In fact, for every cone $ \mathcal C \subset \mathcal L $ with non-empty interior, there exists a Zariski dense subgroup $ \Gamma ' < \Gamma $ with $ \mathcal L_{\Gamma '} \subset \mathcal C $ (see [1]); therefore, we have

$$ \begin{align*} \mathfrak a = \overline{\langle \unicode{x3bb}(\Gamma) \cap \operatorname{int} \mathcal L \rangle}. \end{align*} $$

It is convenient to use a characterization of horospherical limit points in terms of the Busemann function.

Lemma 3.3. For $ \xi \in \Lambda $ , we have $\xi \in \Lambda _h$ if and only if there exists a closed cone ${\mathcal C \subset {\operatorname {int} \mathcal L} \cup \{0\}}$ and a sequence $ \gamma _j \in \Gamma $ satisfying

(3.1) $$ \begin{align} \beta_\xi(o,\gamma_jo) \to \infty \quad \text{and} \quad \beta_\xi(o,\gamma_jo) \in \mathcal C \quad\text{for all large } j\ge 1. \end{align} $$

Proof. Let $\xi =gP\in \Lambda _h$ be as defined in Definition 3.1. Then there exists ${\gamma _j= gp n_j a_{u_j}k_j\in \Gamma }$ for some $p\in P$ , $n_j\in N$ , $k_j\in K$ , and $u_j\to \infty $ in some closed cone $\mathcal C$ contained in $\operatorname {int}\mathcal L\cup \{0\}$ . Fix some closed cone $\mathcal C'\subset \operatorname {int} \mathcal L\cup \{0\}$ whose interior contains  $\mathcal C$ . Note that

$$ \begin{align*} \beta_{\xi}(o, \gamma_j o) &= \beta_{gP}(e, g)+ \beta_{gP}(g, gpn_j a_{u_j})\\ &= \beta_P(g^{-1}, e)+ \beta_P(e,p)+ \beta_P(e, n_j)+\beta_P(e, a_{u_j})\\ & =\beta_{P}(g^{-1},p)+u_j. \end{align*} $$

Therefore, the sequence $\beta _{\xi }(o, \gamma _j)-u_j$ is uniformly bounded. Since $u_j\in \mathcal C$ , $\beta _{\xi }(o,\gamma _j o)\in \mathcal C'$ for all large j. Therefore, equation (3.1) holds. For the other direction, let $\gamma _j$ and $\mathcal C$ satisfy equation (3.1) for $\xi =gP$ for $g\in G$ . Since $G=gNAK$ , we may write $\gamma _j= gn_ja_{u_j} k_j $ for some $n_j\in N, u_j\in \mathfrak a$ and $k_j\in K$ . By a similar computation as above, the sequence $\beta _\xi (o, \gamma _j o)-u_j$ is uniformly bounded. It follows that $u_j\in \mathcal C'$ for all large j and $u_j\to \infty $ . Therefore, for any $T>1$ , there exists $j>1$ such that $\gamma _j (o)\in g N \exp (\mathcal C'-\mathcal C^{\prime }_T)(o)$ . This proves $\xi \in \Lambda _h$ .

We note that the condition in equation (3.1) is independent of the choice of basepoint $ o $ . Indeed, for any $g\in G$ and $\xi \in \mathcal F$ and for all $ \gamma \in \Gamma $ , we have

$$ \begin{align*} \beta_\xi(o, \gamma o) = \beta_\xi(o, go)+\beta_\xi (go, \gamma go)+\beta_\xi (\gamma go, \gamma o) ,\end{align*} $$

and hence

$$ \begin{align*} \|\beta_\xi(o, \gamma o)-\beta_\xi (go, \gamma go)\|&=\|\beta_\xi(o, go)+\beta_\xi (\gamma go, \gamma o)\|\\ &=\|\beta_\xi(o, go)-\beta_{\gamma^{-1}\xi} (o,go)\| \\ & \le 2 \cdot \max_{\eta \in \mathcal{F}} \|\beta_\eta (o, go)\|. \end{align*} $$

Since this bound is independent of $\gamma \in \Gamma $ , the condition in equation (3.1) implies that for any $p=go \in X$ ,

(3.2) $$ \begin{align} \beta_\xi(p,\gamma_j p) \to \infty \quad \text{and} \quad \beta_\xi(p,\gamma_jp) \in \mathcal C \quad\text{for all large }j. \end{align} $$

Let us now consider the following seemingly stronger condition for a limit point being horospherical.

Definition 3.4. For $u\in \mathfrak a^+$ , a point $ \xi \in \mathcal F$ is called a $ u $ -horospherical limit point if for some $p\in X$ (and hence for any $p\in X$ ), there exists a constant $ R>0 $ and a sequence $ \gamma _j \in \Gamma $ satisfying

$$ \begin{align*} \beta_\xi(p,\gamma_jp)\to \infty \quad \text{and}\quad \|\beta_\xi(p,\gamma_jp)-\mathbb{R}_+u\| < R \quad \text{for all }j. \end{align*} $$

We denote the set of $ u $ -horospherical limit points by $ \Lambda _h(u) $ .

By $ G $ -invariance of the Busemann map, the set of horospherical (respectively u-horospherical) limit points is $\Gamma $ -invariant. Therefore, for $x=[g]\in \Gamma \backslash G$ , we may say $x^+:=\Gamma gP$ horospherical (respectively u-horospherical) if $g^+$ is.

For $u\in \mathfrak a$ , we call $x\in \Gamma \backslash G$ a u-periodic point if $x a_u = xm_0$ for some $m_0\in M$ ; note that $xa_{\mathbb R u} M_0$ is then compact. Note that for $u\in \operatorname {int}\mathfrak a^+$ , the existence of a u-periodic point is equivalent to the condition that $u\in \unicode{x3bb} (\Gamma )$ .

Lemma 3.5. Let $u\in \mathfrak a^+$ . If $x\in \Gamma \backslash G$ is u-periodic, then $x^+\in \mathcal F$ is a u-horospherical limit point.

Proof. Since x is u-periodic, there exist $g\in G$ with $x=[g]$ and $\gamma \in \Gamma $ such that $\gamma =g a_u m g^{-1}$ for some $m\in M$ , and $y_\gamma =g^+ \in \Lambda $ . Moreover, for any $k\ge 1$ ,

$$ \begin{align*}\beta_{gP} (go, \gamma^k go)=\beta_{P}(o, a^k_u o)=k u.\end{align*} $$

This implies $gP$ is u-horospherical.

Proposition 3.6. Let $x\in \Gamma \backslash G$ . If $ x^+$ is u-horospherical for some $u\in \unicode{x3bb} (\Gamma )$ , then the closure $\overline {xN}$ contains a $ u $ -periodic point.

Proof. Choose $g\in G$ so that $x=[g]$ . We may assume without loss of generality that $g=k\in K$ , since $kan N= k N a$ , and a translate of a u-periodic point by an element of A is again a u-periodic point. Since $u\in \unicode{x3bb} (\Gamma )$ , there exists a u-periodic point, say, $x_0\in \Gamma \backslash G$ . It suffices to show that

(3.3) $$ \begin{align} \overline{[k]N}\cap x_0 AM\ne \emptyset\end{align} $$

as every point in $x_0AM$ is u-periodic.

Since $k^+$ is $ u $ -horospherical and using equation (2.4), there exists $ R>0 $ and sequences $ \gamma _j \in \Gamma $ , $ u_j \to \infty $ in $ \mathfrak a^+ $ and $ k_j \in K $ and $ n_j \in N $ satisfying $\gamma _j^{-1} k = k_j a_{-u_j} n_j $ or

(3.4) $$ \begin{align} k_j = \gamma_j^{-1} k n_j^{-1} a_{u_j}, \end{align} $$

with $ \|\mathbb {R}_+u - u_j \| < R $ for all $ j $ . Let $ \ell _j \to \infty $ be a sequence of integers satisfying

(3.5) $$ \begin{align} \|\ell_ju - u_j \| < R+\|u\| \;\quad\text{for all} j \ge 1. \end{align} $$

By passing to a subsequence, we may assume without loss of generality that $ \gamma _j^{-1}kP$ converges to some $\xi _0 \in \mathcal {F} $ . Since $\check {N}P$ is Zariski open and $ \Gamma $ is Zariski dense, we may choose $g_0\in G$ such that $x_0=[g_0]$ and $g_0^{-1}\xi _0\in \check {N}P$ . Let $h_0\in \check {N}$ be such that $ \xi _0=g_0 h_0 P $ . Since $ g_0\check {N}P $ is open and $\gamma _j^{-1}kP\to g_0h_0P$ , we may assume that for all j, there exists $ h_j \in \check {N} $ satisfying $ g_0 h_j P = \gamma _j^{-1}k P = k_j P $ with $ h_j \to h_0 $ . Let $p_j= a_{v_j}m_j \tilde n _j \in P=AMN$ be such that $g_0 h_j p_j= k_j$ ; since $h_j\to h_0$ and the product map $\check {N}\times P\to \check {N}P$ is a diffeomorphism, the sequence $p_j$ , as well as ${v_j}\in \mathfrak a$ , are bounded.

Therefore, by equation (3.4), we get for all j,

$$ \begin{align*} g_0 &= k_j p_j^{-1}h_j^{-1} \\& =\gamma_j^{-1} kn^{-1}_j a_{u_j} ( \tilde n^{-1}_j m_j^{-1} a_{-v_j}) h_j^{-1} \\ &= \gamma_j^{-1} k n^{-1}_j (a_{u_j} \tilde n^{-1}_j a_{-u_j}) a_{u_j} m_j^{-1} a_{-v_j}h_j^{-1} \\ &= \gamma_j^{-1} k n^{-1}_j (a_{u_j} \tilde n^{-1}_j a_{-u_j}) m_j^{-1} (a_{u_j-v_j}h_j^{-1} a_{-u_j+v_j}) a_{u_j-v_j}. \end{align*} $$

Since $h_j^{-1} \in \check {N}$ and $ v_j \in \mathfrak a $ are uniformly bounded and since $u_j\to \infty $ within a bounded neighborhood of the ray $\mathbb R_+ u\in \operatorname {int}\mathfrak a^+$ , we have

$$ \begin{align*} {\tilde h}_j = a_{u_j-v_j}h_j^{-1} a_{-u_j+v_j} \to e \quad \text{in }\check{N}. \end{align*} $$

By setting $ n^{\prime }_j = n^{-1}_j (a_{u_j} \tilde n ^{-1}_j a_{-u_j}) \in N $ , we may now write

$$ \begin{align*} g_0 = \gamma_j^{-1} k n^{\prime}_j m_j^{-1} {\tilde h}_j a_{u_j-v_j}. \end{align*} $$

Since $x_0$ is u-periodic, there exists $ \gamma _0 \in \Gamma $ such that $ \gamma _0 = g_0 a_u m_0 g_0^{-1} $ for some $ m_0 \in M $ . Hence, for all $j\ge 1$ ,

$$ \begin{align*} \gamma_0^{-\ell_j} &= g_0 a_{-\ell_j u} m_0^{-\ell_j} g_0^{-1} = (\gamma_j^{-1} kn^{\prime}_j m_j^{-1} {\tilde h}_j a_{u_j-v_j} ) ( a_{-\ell_j u} m_0^{-\ell_j }) g_0^{-1}. \end{align*} $$

In other words,

$$ \begin{align*} \gamma_j^{-1} k n^{\prime}_j = \gamma_0^{-\ell_j}g_0 m_0^{\ell_j} a_{-u_j+\ell_j u+v_j} {\tilde h}_j^{-1}m_j. \end{align*} $$

Since the sequence $ -u_j+\ell _j u+ v_j \in \mathfrak a $ is uniformly bounded by equation (3.5) and $ {\tilde h}_j \to e $ in $ \check {N} $ , we conclude that the sequence $ \Gamma k n^{\prime }_j $ has an accumulation point in $ \Gamma g_0 AM $ . This proves equation (3.3).

It turns out that a horospherical limit point is also $ u $ -horospherical for any $u\in \operatorname {int} \mathcal L$ .

Proposition 3.7. For each $ u \in {\operatorname {int} \mathcal L} $ , we have $ \Lambda _h=\Lambda _h(u) $ .

Proof. Let $ \xi \in \Lambda _h $ . By definition, there is a sequence $ \gamma _j \in \Gamma $ satisfying $v_j:=\beta _\xi (e,\gamma _j)\to \infty $ with the sequence $\|v_j\|^{-1}v_j$ converging to some point $v_0\in \operatorname {int} \mathcal L$ . By passing to a subsequence, we may assume that $\gamma _j^{-1} \xi $ converges to some $\xi _0\in \mathcal F$ .

Let $u\in \operatorname {int}\mathcal L$ . We claim that $\xi \in \Lambda _h(u)$ . We first consider the case $u\not \in \mathbb R_+ v_0$ . Let $r:=\operatorname {rank} G-1\ge 0$ . Since $\bigcup _{\gamma \in \Gamma }\mathbb R_+\unicode{x3bb} (\gamma )$ is dense in $\mathcal L$ , there exist $w_1, \ldots , w_r\in \unicode{x3bb} (\Gamma )$ such that $v_0$ belongs to the interior of the convex cone spanned by $u, w_1, \ldots , w_r$ , so that

$$ \begin{align*}v_0= c_0 u+ \sum_{\ell=1}^r c_\ell w_\ell \end{align*} $$

for some positive constants $c_0,\ldots , c_\ell $ .

Since $\|v_j\|^{-1}v_j \to v_0$ , we may assume, by passing to a subsequence, that for each $j\ge 1$ , we have

(3.6) $$ \begin{align} \|v_j\|^{-1} v_j = c_{0,j} u +\sum_{\ell =1}^r c_{\ell,j} w_\ell \end{align} $$

for some positive $c_{\ell ,j}$ , $\ell =0, \ldots , r$ . Note that for each $0\le \ell \le r$ , $c_{\ell ,j}\to c_\ell $ as $j\to \infty $ .

By Lemma 2.1, we can find a loxodromic element $g_1\in \Gamma $ and a neighborhood $U_1$ of $\xi _0$ such that $\unicode{x3bb} (g_1^{-1})=w_1$ , $\{y_{g_1}\}\times U_1\subset \mathcal F^{(2)}$ and $g_1^{-k}U_1\to y_{g_1^{-1}}$ uniformly. Applying Lemma 2.1 once more, we can find $ g_2 \in \Gamma $ satisfying $ \unicode{x3bb} (g_2^{-1}) = w_2 $ and a neighborhood $ U_2 \subset \mathcal {F} $ of $ y_{g_1^{-1}} $ satisfying $ \{y_{g_2} \}\times U_2 \subset \mathcal {F}^{(2)} $ and that $ g_2^{-k}U_2 \to y_{g_2^{-1}} $ uniformly.

Continuing inductively, we get elements $ g_1,\ldots ,g_r \in \Gamma $ and open sets $ U_1,\ldots ,U_r \subset \mathcal {F} $ satisfying that for all $ \ell = 1,\ldots ,r $ :

1. (1) $ {w}_\ell =\unicode{x3bb} (g_\ell ^{-1}) $ ;

2. (2) $ y_{g_{\ell -1}^{-1}} \in U_{\ell } $ ;

3. (3) $ g_\ell ^{-k}U_\ell \to y_{g_\ell ^{-1}} $ uniformly; and

4. (4) $ \{ y_{g_\ell } \}\times U_\ell $ is a relatively compact subset of $ \mathcal {F}^{(2)} $ .

We set $\xi _\ell :=y_{g_\ell ^{-1}}$ for each $1\le \ell \le r$ ; so $U_{\ell }$ is a neighborhood of $\xi _{\ell -1}$ for each $1\le \ell \le r$ .

Since $\mathcal Q_{\eta _0}:=\{\eta \in \mathcal F: (\eta _0, \eta )\in \mathcal F^{(2)}\}=\bigcup _{R>0} O_R (\eta _0, o)$ for any $\eta _0\in \mathcal F$ and $U_\ell \subset \mathcal Q_{y_{g_\ell }}$ is a relatively compact subset of $\mathcal {F}^{(2)} $ , there exists $ R_\ell>0 $ such that $ U_\ell \subset O_{R_\ell }(y_{g_\ell },o) $ . Since $g_\ell ^ko $ converges to $y_{g_\ell }$ as $k\to +\infty $ , by Lemma 2.3(2),

(3.7) $$ \begin{align} O_{R_\ell}(y_{g_\ell}o,o) \subset O_{R_\ell+1}(g_\ell^k o,o) \end{align} $$

for all sufficiently large $k>1$ .

For each $1\le \ell \le r$ and $j\ge 1$ , let $k_{\ell ,j}$ be the largest integer smaller than $c_{\ell , j}\|v_j\|$ . As $\|v_j\|\to \infty $ and $c_{\ell ,j}\to c_\ell $ , we have $k_{\ell , j}\to \infty $ as $j\to \infty $ . By the uniform contraction $ g_\ell ^{-k}U_i \to \xi _\ell $ , there exists $j_0>1$ such that for all $j\ge j_0$ ,

(3.8) $$ \begin{align} \gamma_{j}^{-1}\xi \in U_1, \quad g_\ell^{-k_{\ell,j}}U_\ell \subseteq U_{\ell+1} ,\quad \text{and} \quad U_\ell \subset O_{R_\ell+1}(g_\ell^{k_\ell, j}o,o) \end{align} $$

for all $ \ell =1,\ldots ,r $ .

For each $j\ge j_0$ , we now set

$$ \begin{align*} \tilde{\gamma}_j:=\gamma_j g_1^{k_{1,j}} g_2^{k_{2,j}} \ldots g_r^{k_{r,j}} \in \Gamma. \end{align*} $$

We claim that $\beta _\xi (e,\tilde {\gamma }_j)\to \infty $ as $j\to \infty $ and that

(3.9) $$ \begin{align} \sup_{j\ge j_0} \|\beta_\xi(e,\tilde{\gamma}_j)-\mathbb R_+ u \|<\infty ;\end{align} $$

this proves that $\xi $ is u-horospherical.

Fix $ j\ge j_0 $ and for each $1\le \ell \le r$ , let $ k_\ell :=k_{\ell ,j} $ , $b_\ell :=c_{\ell ,j}\|v_j\|$ , and set

$$ \begin{align*} h_\ell = g_1^{k_1} g_2^{k_2} \ldots g_\ell^{k_\ell}, \end{align*} $$

and $ g_0=e $ . The cocycle property of the Busemann function gives that

(3.10) $$ \begin{align} \beta_\xi(e,\tilde{\gamma}_j) =\beta_\xi(e,\gamma_j) -\sum_{\ell=1}^{r} \beta_{h_{\ell-1}^{-1}\gamma_j^{-1}\xi}(g^{k_\ell}_\ell, e). \end{align} $$

By equation (3.8), $\gamma _j^{-1}\xi \in U_1$ and for each $1\le \ell \le r$ ,

$$ \begin{align*}h_{\ell -1}^{-1}\gamma_j^{-1} \xi \in g_\ell^{-k_\ell}\ldots g_1^{-k_1}U_1\subset U_{\ell+1}\subset O_{R_\ell +1}(g_\ell^{k_\ell}o, o ).\end{align*} $$

Hence, by Lemma 2.3(1), there exists $\kappa \ge 1$ such that for each $1\le \ell \le r$ ,

$$ \begin{align*} \| \beta_{h_{\ell-1}^{-1}\gamma_j^{-1}\xi}(g^{k_\ell}_\ell, e) -\mu(g^{-k_\ell}_\ell)\|\le \kappa (R_{\ell}+1).\end{align*} $$

Note that for some $C_\ell>0$ , $\|\mu (g_\ell ^{-k}) -k \unicode{x3bb} (g_\ell ^{-1})\|\le C_\ell $ for all $k\ge 1$ . Since $\unicode{x3bb} (g_\ell ^{-1})=w_\ell $ , we get

$$ \begin{align*} \| \beta_{h_{\ell-1}^{-1}\gamma_j^{-1}\xi}(g^{k_\ell}_\ell, e) -k_\ell w_\ell \|\le \kappa (R_{\ell}+1)+ C_\ell .\end{align*} $$

Therefore, by equation (3.10), we obtain

$$ \begin{align*}\bigg\|\beta_\xi(e, \tilde \gamma_j) -\bigg(v_j -\sum_{\ell=1}^r k_\ell w_\ell \bigg) \bigg\| \le \kappa \sum_{\ell=1}^r (R_\ell + C_\ell+1) .\end{align*} $$

By equation (3.6), we have

$$ \begin{align*}c_{0,j} \|v_j\| u = v_j -\sum_{\ell=1}^{r} b_\ell w_\ell .\end{align*} $$

Since $|b_\ell -k_\ell |\le 1$ and $c_{0,j}>0$ , we deduce that for all $j\ge j_0$ ,

$$ \begin{align*} &\|\beta_\xi(e, \tilde \gamma_j)-\mathbb R_+ u\| \le \| \beta_\xi(e, \tilde \gamma_j)- c_{0,j} \|v_j\|\cdot u \| \\ &\quad\le \bigg\| \beta_\xi(e, \tilde \gamma_j)- \bigg(v_j -\sum_{\ell=1}^r k_\ell w_\ell \bigg) \bigg\| + \sum_{\ell=1}^r \| k_\ell w_\ell -b_\ell w_\ell \| \\&\quad\le \kappa \sum_{\ell=1}^r (R_\ell + C_\ell +\|w_\ell\| +1). \end{align*} $$

This proves equation (3.9) and, consequently, $\xi $ is u-horospherical for any $u\notin \mathbb R_+ v_0.$ To show that $\xi $ is $v_0$ -horospherical, fix any $u\notin \mathbb R_+v_0$ and $\tilde \gamma _j\in \Gamma $ be a sequence as in equation (3.9) associated to u. If we set $\tilde v_j=\beta _\xi (e, \tilde \gamma _j)$ , then $\|\tilde v_j\|^{-1} \tilde v_j$ converges to a unit vector in $\operatorname {int} \mathcal L$ proportional to u. Therefore, by repeating the same argument only now switching the roles of $v_0$ and u, we prove that $\xi $ is $v_0$ -horospherical as well. This completes the proof.

We may now prove Theorem 3.2.

Proof of Theorem 3.2

Let $g\in G$ be such that $\xi =g^+\in \Lambda $ is a horospherical limit point. Set $Y:=\overline {[g]NM}$ . We claim that $Y=\mathcal E$ . By Benoist [1], the group generated by $\unicode{x3bb} (\Gamma )\cap \operatorname {int} \mathcal L$ is dense in $\mathfrak a$ . Hence, for every $ \varepsilon> 0 $ , there exist loxodromic elements $ \gamma _1,\ldots ,\gamma _{q} \in \Gamma $ such that

$$ \begin{align*} \unicode{x3bb}(\gamma_1),\ldots,\unicode{x3bb}(\gamma_{q}) \in \mathrm{Int}\mathcal L \end{align*} $$

and the group $\mathbb Z\unicode{x3bb} (\gamma _1)+\cdots +\mathbb Z \unicode{x3bb} (\gamma _{q})$ is an $\varepsilon $ -net in $\mathfrak a$ , that is, its $\varepsilon $ -neighborhood covers all $\mathfrak a$ . Denote $u_i = \unicode{x3bb} (\gamma _i) $ for $ i=1,\ldots ,q $ . By Proposition 3.7, the point $ \xi $ is $u_1 $ -horospherical. By Proposition 3.6, there exists a $ u_1 $ -periodic point $x_1 \in \mathcal E $ contained in $ Y $ , set

$$ \begin{align*} Y_1: = \overline{x_1 NM} \subset Y. \end{align*} $$

By Lemma 3.5, $x_1^+$ is $u_1$ -horospherical; in particular, it is a horospherical limit point. Therefore, we can inductively find a $u_i$ -periodic point $ x_i $ in $ Y_{i-1}=\overline {x_{i-1} NM} $ for each $2\le i\le q$ . By periodicity, $x_i (\exp {u_i}) M=x_i M$ , and hence $Y_i \exp {\mathbb Z u_i} = Y_i$ for each $1\le i\le q$ . Therefore, we obtain

$$ \begin{align*}Y\supset Y_1\, {\exp {\mathbb Z u_1}}\supset Y_2\, {\exp ({\mathbb Z u_1+\mathbb Z u_2})}\supset \cdots \supset Y_q\, {\exp {\bigg(\sum_{i=1}^q \mathbb Z u_i}\bigg)}.\end{align*} $$

Recalling the dependence of $Y_q$ and $\sum _{i=1}^q \mathbb Z u_i$ on $\varepsilon $ , set

$$ \begin{align*}Z_\varepsilon:= Y_q MN \exp \bigg(\sum_{i=1}^q \mathbb Z u_i\bigg) \subset Y.\end{align*} $$

Since $MN \exp (\sum _{i=1}^q \mathbb Z u_i)$ is an $\varepsilon $ -net of P and $\mathcal E$ is P-minimal, $Z_\varepsilon $ is a $2\varepsilon $ -net of $\mathcal E$ for all $\varepsilon>0$ . Since Y contains a $2\varepsilon $ -net of $\mathcal E$ for all $\varepsilon>0$ and Y is closed, it follows that $Y=\mathcal E$ .

For the other direction, suppose that $[g]NM$ is dense in $\mathcal E$ for $g\in G$ . Choose any $u\in \operatorname {int} \mathcal L$ and a closed cone $\mathcal C \subset \operatorname {int} \mathcal L\cup \{0\}$ which contains u. Then $\mathcal H_\xi =gN(\exp \mathcal C )(o)$ is a $\Gamma $ -tight horoball. Let $t>1$ . Since $ga_{-2t u}\in \mathcal E$ , there exist $\gamma _i\in \Gamma $ , $n_i\in N$ , $m_i\in M$ , and $q_i\to e$ in G such that for all $i\ge 1$ , $\gamma _i g n_i m_i q_i= g a_{-2t u}$ . Since $d(\gamma _i^{-1}g, gn_i m_ia_{2t u})\le d(q_i a_{2t u},a_{2t u})\to 0 $ as $i\to \infty $ , it follows that for all sufficiently large $i\ge 1$ , $\gamma _i^{-1} go \in \mathcal H_\xi (t)$ . Hence, $g^+$ is a horospherical limit point by Definition 3.1.

4 Topological mixing and directional limit points

There is a close connection between denseness of $ N $ -orbits and the topological mixing of one-parameter diagonal flows with direction in $ \operatorname {int} \mathcal L $ . This connection allows us to make use of recent topological mixing results by Chow and Sarkar [3]: recall the notation $\Omega _0$ from equation (2.2).

Theorem 4.1. [3]

For any $u\in \operatorname {int} \mathcal L$ , $\{a_{tu} : t\in \mathbb R\}$ is topologically mixing on $\Omega _0$ , that is, for any open subsets $\mathcal O_1, \mathcal O_2$ of $\Gamma \backslash G$ intersecting $\Omega _0$ ,

$$ \begin{align*}\mathcal O_1\exp tu \cap \mathcal O_2\ne \emptyset \quad\text{ for all large}\ |t|\gg 1.\end{align*} $$

The above theorem was predated by a result of Dang [6] in the case where $ M $ is abelian.

4.1 N-orbits based at directional limit points along $\operatorname {int} \mathcal L$

Definition 4.2. For $u\in \operatorname {int} \mathfrak a^+$ , denote by $\Lambda _u$ the set of all u-directional limit points, that is, $\xi \in \Lambda _u$ if and only if $\limsup _{t\to +\infty } \Gamma g \exp (tu)\ne \emptyset $ for some (and hence any) $g\in G$ with $gP=\xi $ .

It is easy to see that $\Lambda _u\subset \Lambda $ for $u\in \operatorname {int} \mathfrak a^+$ .

Proposition 4.3. If $[g]\in \mathcal E_0$ satisfies $g^+\in \Lambda _u$ for some $u\in \operatorname {int}\mathcal L$ , then

$$ \begin{align*}\overline{[g]N}=\mathcal E_0.\end{align*} $$

Proof. Since $\Omega _0 N=\mathcal E_0$ , we may assume without loss of generality that $x=[g]\in \Omega _0$ . There exist $\gamma _i\in \Gamma $ and $t_i\to +\infty $ such that $\gamma _i g a_{t_iu} $ converges to some $h\in G$ . In particular, $x \exp (t_i u)\to [h]$ . Since $x a_{t_iu}\in \Omega _0$ and $\Omega _0$ is $ A $ -invariant and closed, we have $[h]\in \Omega _0$ . We write $\gamma _i g a_{t_iu}=hq_i$ , where $q_i\to e$ in G. Therefore, $xN=[h] q_i N a_{-t_iu}$ for all $i\ge 1$ . Let $\mathcal O\subset \Gamma \backslash G$ be any open subset intersecting $\Omega _0$ . It suffices to show that $xN\cap \mathcal O\ne \emptyset $ . Let $\mathcal O_1$ be an open subset intersecting $\Omega _0$ and $V\subset \check {P}$ be an open symmetric neighborhood of e such that $\mathcal O_1 V\subset \mathcal O$ .

Since $q_i\to e$ and $NV$ is an open neighborhood of e in G, there exists an open neighborhood, say, U of e in G and $i_0$ such that $U\subset q_i NV$ for all $i\ge i_0$ . By Theorem 4.1, we can choose $i>i_0$ such that $[h] U\kern1.3pt{\cap}\kern1.3pt \mathcal O_1 a_{t_iu}\kern1.3pt{\ne}\kern1.3pt \emptyset $ . It follows that ${[h] q_i NV a_{-t_iu} \kern1.3pt{\cap}\kern1.3pt \mathcal O_1\kern1.3pt{\ne}\kern1.3pt \emptyset }$ . Since $V\subset a_{-t_iu} V a_{t_iu}$ as $u\in \mathfrak a^+$ , we have

$$ \begin{align*}[h] q_i NV a_{-t_iu} \cap \mathcal O_1 \subset [h] q_i N a_{-t_iu} V \cap \mathcal O_1 .\end{align*} $$

Since $V=V^{-1}$ , we get $[h] q_i N a_{-t_iu} \cap \mathcal O_1 V \ne \emptyset $ . Therefore, $xN\cap \mathcal O\ne \emptyset $ , as desired.

This immediately implies the following corollary.

Corollary 4.4. If $[g]\in \Omega _0$ is u-periodic for some $u\in \operatorname {int}\mathcal L$ , then

$$ \begin{align*}\overline{[g]N}=\mathcal E_0.\end{align*} $$

Proof. Since $ [g] (\exp ku) =[g]m_0^k$ for any integer k and M is compact, we have $g^+\in \Lambda _u$ . Therefore, the claim follows from Proposition 4.3.

We may now conclude our main theorem in its fullest form.

Theorem 4.5. Let $[g]\in \mathcal E_0$ . The following are equivalent:

1. (1) $g^+\in \Lambda $ is a horospherical limit point;

2. (2) $[g]N$ is dense in $\mathcal E_0$ ;

3. (3) $[g]NM$ is dense in $\mathcal E$ .

Proof. The implication $(2)\Rightarrow (3)$ is trivial and $ (3) \Rightarrow (1) $ was shown in Theorem 3.2. Hence, let us prove $ (1)\Rightarrow (2) $ .

Let $x=[g]\in \mathcal E_0$ . Suppose that $g^+\in \Lambda _h$ . Fix any $u\in \unicode{x3bb} (\Gamma )\cap \operatorname {int} \mathcal L_\Gamma $ . By Propositions 3.7 and 3.6, $xN$ contains a u-periodic point, say, $x_0$ . Hence, by Corollary 4.4, $\overline {xN}\supset \overline {x_0N}\supset \Omega _0 N=\mathcal E_0$ . This proves $(1)\Rightarrow (2)$ .

5 Conical limit points, minimality, and Jordan projection

A point $\xi \in \mathcal F$ is called a conical limit point of $\Gamma $ if there exists a sequence $ u_j \to \infty $ in $ \mathfrak a^+ $ such that for some (and hence every) $ g \in G $ with $ \xi = gP $ ,

$$ \begin{align*} \limsup_{j\to \infty} \Gamma g a_{u_j} \neq \emptyset. \end{align*} $$

A conical limit point of $\Gamma $ is indeed contained in $\Lambda $ . We consider the following restricted notion.

Definition 5.1. We call $\xi \in \mathcal F$ a strongly conical limit point of $\Gamma $ if there exists a closed cone $ \mathcal C \subset \operatorname {int} \mathcal L \cup \{0\} $ and a sequence $ u_j \to \infty $ in $ \mathcal C $ such that for some (and hence every) $ g \in G $ with $ \xi = gP $ ,

$$ \begin{align*} \limsup_{j\to \infty} \Gamma g a_{u_j} \neq \emptyset. \end{align*} $$

Remarks 5.2. We mention that a conical limit point defined in [4] for $\Gamma <\operatorname {SL}_d(\mathbb R)$ coincides with our strongly conical limit point.

Lemma 5.3. Any strongly conical limit point of $\Gamma $ is horospherical.

Proof. Suppose that $ \xi = gP $ is strongly conical, that is, there exist $\gamma _j\in \Gamma $ and $ u_j \to \infty $ in some closed cone $\mathcal C\subset \operatorname {int} \mathcal L \cup \{0\}$ such that $\gamma _j g a_{u_j}$ converges to some $h\in G$ . Write $\gamma _j g a_{u_j}=hq_j$ , where $q_j\to e$ in G. Let $\mathcal C'$ be a closed cone contained in $\operatorname {int} \mathcal L \cup \{0\}$ whose interior contains $\mathcal C \smallsetminus \{0\}$ .

Then $\gamma _j^{-1}=g a_{u_j}q_j^{-1} h^{-1}$ and

$$ \begin{align*}\beta_{gP} (e, \gamma_j^{-1} ) =\beta_P (g^{-1}, a_{u_j}q_j^{-1} h^{-1})= \beta_{P} (g^{-1}, q_j^{-1}h^{-1}) +\beta_P(e, a_{u_j}) .\end{align*} $$

Since $\beta _P(e, a_{u_j})=u_j$ and $q_j^{-1}h^{-1}$ are uniformly bounded, the sequence

$$ \begin{align*}\beta_{gP}(e, \gamma_j^{-1}) - u_j\end{align*} $$

is uniformly bounded. Since $u_j\in \mathcal C$ and $\mathcal C\subset \operatorname {int}\mathcal C'\cup \{0\}$ , it follows that

$$ \begin{align*}\beta_{gP}(e, \gamma_j^{-1})\in \mathcal C'\end{align*} $$

for all sufficiently large j. This proves that $\xi \in \Lambda _h$ .

Corollary 5.4. For any $g\in G$ with strongly conical $g^+\in \mathcal F$ , we have

$$ \begin{align*}\overline{[g] NM}=\mathcal E .\end{align*} $$

5.1 Directionally conical limit points

If $v\in {\operatorname {int} \mathcal L}$ , then clearly $\Lambda _v$ is contained in the horospherical limit set of $\Gamma $ , and hence any $NM$ -orbit based at a point of $\Lambda _v$ is dense in $\mathcal E$ . However, we would like to show in this section that the existence of a point in $\Lambda _v$ for $v\in \partial \mathcal L_\Gamma $ implies the existence of a non-dense $NM$ -orbit in $\mathcal E$ .

The flow $\exp (\mathbb R u)$ is said to be topologically transitive on $\Omega /M=\{\Gamma g M : g^{\pm }\in \Lambda \}$ if for any open subsets $\mathcal O_1, \mathcal O_2$ intersecting $\Omega /M$ , there exists a sequence $t_n \to +\infty $ such that $\mathcal O_1\cap \mathcal O_2 a_{t_n u}\ne \emptyset $ .

We make the following simple observation.

Lemma 5.5. For $g\in \Omega $ , we have

$$ \begin{align*}\overline{gNM}\supset \Omega\quad \text{if and only if}\, \ \overline {gw_0\check{N}M}\supset \Omega.\end{align*} $$

Proof. We have $\check {N}=w_0 N w_0^{-1}$ . Note that $[g]\in \Omega $ if and only if $[gw_0]\in \Omega $ , since $(g w_0)^{\pm }=g^{\mp }$ . So $\Omega w_0=\Omega $ . Hence, $gNM$ is dense in $\Omega $ if and only if $gw_0 \check {N}M w_0^{-1} $ is dense in $\Omega $ if and only if $[g]w_0\check {N}M$ is dense in $\Omega w_0=\Omega $ .

Since the opposition involution preserves $\mathcal L$ and $\unicode{x3bb} (g^{-1})=\operatorname {i}\unicode{x3bb} (g)$ for any loxodromic element, it follows that $\unicode{x3bb} (\gamma )\in \partial \mathcal L$ if and only if $\unicode{x3bb} (\gamma ^{-1})\in \partial \mathcal L$ .

Proposition 5.6

1. (1) If $\Lambda =\Lambda _h$ , then $\exp (\mathbb R v)$ is topologically transitive on $\Omega /M$ for any $v\in \operatorname {int}\mathfrak a^+$ such that $\Lambda _v\ne \emptyset $ .

2. (2) For any loxodromic element $\gamma \in \Gamma $ with $\{y_{\gamma }, y_{\gamma ^{-1}}\}\subset \Lambda _h$ , the flow $\exp (\mathbb R \unicode{x3bb} (\gamma ))$ is topologically transitive on $\Omega /M$ .

Proof. Assume that $\Lambda =\Lambda _h$ ; so the $NM$ -action on $\mathcal E$ is minimal. Suppose that $ \Lambda _v\ne \emptyset $ for some $v\in \operatorname {int}\mathfrak a^+.$ We claim that for any ${\mathcal O}_1, {\mathcal O}_2$ be two right M-invariant open subsets intersecting $\Omega $ , ${\mathcal O}_1 \exp (t_iv) \cap {\mathcal O}_2\ne \emptyset $ for some sequence $t_i \to +\infty $ . Choose $x=[g]\in \Omega $ so that $g^+\in \Lambda _v$ . Then there exists $\gamma _i\in \Gamma $ and $t_i\to +\infty $ such that $\gamma _i g a_{t_i v}$ converges to some $g_0$ . Note that $x_0:=[g_0]\in \Omega $ . So write $\gamma _i g a_{t_i v}= g_0 h_i$ with $h_i\to e$ . By the $NM$ -minimality assumption, $x NM $ intersects every open subset of $\Omega $ . Since $v\in \operatorname {int} \mathfrak a^+$ and hence $a_{-tv} n a_{tv}\to e$ as $t\to +\infty $ , we may assume without loss of generality that $x \in {\mathcal O}_1$ . Choose an open neighborhood U of e in G so that ${\mathcal O}_1\supset x U M$ . Note that there exists a sequence $T_i\to \infty $ as $i\to \infty $ such that for all i,

$$ \begin{align*}x U M a_{t_i v} \supset x a_{t_i v} a_{-t_i v} \check{N}_\varepsilon M a_{t_i v} \supset x_0 h_i \check{N}_{T_i}, \end{align*} $$

where $ \check {N}_{R}=\check {N} \cap B_R^G $ is the set of elements of $ \check {N} $ of norm $ \leq R $ . So ${\mathcal O}_1 a_{t_i v}\supset x_0h_i \check {N}_{T_i}$ .

Choose an open neighborhood V of e in G and some open subset ${\mathcal O}_2'$ intersecting $\Omega $ so that ${\mathcal O}_2\supset {\mathcal O}_2' V$ . Since $x_0\check {N}M$ is dense in $\Omega $ , $x_0 n\in {\mathcal O}_2'$ for some $n\in \check {N}$ . Hence, $x_0 h_i n = x_0 n (n^{-1} h_i n)\in {\mathcal O}_2' V\subset {\mathcal O}_2 $ for all i large enough so that $n^{-1} h_i n\in V$ . Therefore, for all i such that $n\in \check {N}_{T_i}$ , we get

$$ \begin{align*}x_0h_in\in {\mathcal O}_1 a_{t_i v}\cap {\mathcal O}_2\ne \emptyset.\end{align*} $$

This proves the first claim.

Now suppose that $\gamma \in \Gamma $ is a loxodromic element with $y_{\gamma }, y_{\gamma ^{-1}}\in \Lambda _h$ . Write $\gamma =g m a_{ v} g^{-1} $ for some $g\in G$ and $m\in M$ . Since $y_\gamma =g^+$ and $y_{\gamma ^{-1}}=gw_0^+,$ we have each $[g]NM$ and $[g]w_0 NM$ contains $\Omega $ in its closure. Now in the notation of the proof of the first claim, note that $x_0=[g_0]\in [g]M$ since $[g]\exp (\mathbb R v) M$ is closed. Therefore, each $\overline {x_0 NM}$ and $\overline {x_0 \check {N}M}$ contains $ \Omega $ . Based on this, the same argument as above shows the topological transitivity of $\exp \mathbb R v$ , which finishes the proof since $v=\unicode{x3bb} (\gamma )$ .

Since $\mathcal L$ is invariant under the opposition involution $\operatorname {i}$ and $\unicode{x3bb} (\gamma )=\operatorname {i} \unicode{x3bb} (\gamma ^{-1})$ for any loxodromic element $\gamma \in \Gamma $ , the Jordan projection $\unicode{x3bb} (\gamma )$ belongs to $ \partial \mathcal L$ if and only if the Jordan projection $\unicode{x3bb} (\gamma ^{-1})$ belongs to $ \partial \mathcal L$ . Together with the result of Dang and Gloriuex [7, Proposition 4.7], which says that $\exp (\mathbb R u)$ is not topologically transitive on $\Omega /M$ for any $u\in \partial \mathcal L\cap \operatorname {int} \mathfrak a^+$ , Proposition 5.6 implies the following corollary.

Corollary 5.7

1. (1) If $\Lambda _v\ne \emptyset $ for some $v\in \partial \mathcal L\cap \operatorname {int} \mathfrak a^+$ , then

   $$ \begin{align*}\Lambda\ne \Lambda_h.\end{align*} $$

2. (2) For any loxodromic element $\gamma \in \Gamma $ , we have $\unicode{x3bb} (\gamma )\in \partial \mathcal L$ if and only if

   $$ \begin{align*}\{y_\gamma, y_{\gamma^{-1}}\} \not\subset \Lambda_h .\end{align*} $$
   Hence, if $\Lambda =\Lambda _h$ , then $\unicode{x3bb} (\Gamma )\subset \operatorname {int} \mathcal L$ .