PROXIMALITY AND EQUIDISTRIBUTION ON THE FURSTENBERG BOUNDARY

A. GORODNIK AND F. MAUCOURANT

Abstract. Let G be a connected semisimple Lie group with finite center and without compact factors, P a minimal parabolic subgroup of G, and Γ a lattice in G. We prove that every Γ-orbits in the Furstenberg boundary G/P is equidistributed for the averages over Riemannian balls. The proof is based on the proximality of the action of Γ on G/P.

1. Introduction

Let G be a connected semisimple Lie group with finite center and without compact factor, and Γ a lattice in G, that is, a discrete subgroup of G such that Γ\G has finite volume. In this article we investigate the distribution of orbits of Γ acting on the Furstenberg boundary of G. Recall that the Furstenberg boundary can be identified with the factor space G/P, where P is a minimal parabolic subgroup of G. It is known that every orbit of Γ in G/P is dense (see [Mo]). We show that orbits of Γ are equidistributed with respect to the averages over Riemannian balls.

Since we study the action of a nonamenable group on a space without a finite invariant measure, our result lies outside the scope of the classical ergodic theory. The published results about distribution of dense orbits of nonamenable groups are limited to a few special examples. Arnold and Krylov showed in [AK] that dense orbits of groups generated by two rotations acting on the 2-dimensional sphere are equidistributed. A similar problem was considered by Kazhdan in [Ka] where he studied the action of a group generated by two affine isometries on the plane ℝ^{2}. Distribution of dense orbits of a lattice in SL(2, ℝ) acting on ℝ^{2} was investigated by Ledrappier [L] and Nogueira [N].

Let X be the symmetric space of G equipped with a right invariant Riemannian metric d. Note that X can be identified with L\G for a maximal compact subgroup L of G.

Fix x, x̃ ∈ X and denote by K and K̃ the stabilizers of x and x̃ respectively. Let ν and ν̃ be the probability Haar measures on K and K̃ and mx̃ the harmonic measures at x̃ on G/P, that is, the unique K̃-invariant probability measure on G/P. For S ⊂ G and T > 0, define

The first author is partially supported by NSF grant 0400631.

S_{T} (x̃) = {s ∈ S : d(x, x̃s) < T },
S_{T} = S_{T} (x).

Our main result is the following theorem. Theorem 1. For every f ∈ C(G/P), x̃ ∈ X, and y ∈ G/P,

lim_{T →∞} \frac{1}{|Γ_{T} (x̃)|} ∑_{γ∈Γ_{T} (x̃)} f(γy) = ∫_{G/P} fdmx̃,

Moreover, the convergence is uniform for y ∈ G/P.

We remark that it was shown in [EM] (see also [DRS]) that

(1)  |Γ_{T} (x̃)| ∼_{T →∞} \frac{Vol(G_{T} (x̃))}{Vol(Γ\G)},

and the exact asymptotics of the volume Vol(G_{T} (x̃)) = Vol(G_{T}) as T → ∞ was computed in [Kn].

The first result in the direction of Theorem 1 was established in [Ma], where the case of the real hyperbolic spaces was considered. A similar problem for G = SL(n, ℝ) was considered in [G], where the averages were defined with respect to a matrix norm. A different proof of Theorem 1 is given in [GO]. The approach presented here allows us to show that the convergence is uniform whereas in [GO], the uniformness doesn’t seem obvious from the method of the proof. While the proof in [GO] uses equidistribution of solvable flows on Γ\G, our proof is based on the strong proximality of the action of G on G/P (see Theorem 2 below). This result is of independent interest, and it might be useful for other applications.

Recall that an action of a group H on a compact metric space (Y, d) is called proximal if for every u, v ∈ Y there exists a sequence {h_{n}} ⊂ H such that d(h_{n}u, h_{n}v) → 0 as n → ∞. The fact that the action of G on G/P is proximal plays important role in the study of random walks on G (see, for example, [F]). It turns out that a typical sequence in G acts on G/P in proximal fashion. Theorem 2 (Strong proximality). Let O be a neighborhood of the diagonal in G/P × G/P and u, v ∈ G/P. Then

lim_{T →∞} \frac{Vol(\{g ∈ G_{T} (x̃) : (gu, gv) ∉ O\})}{Vol(G_{T} (x̃))} = 0

and

lim_{T →∞} \frac{|\{γ ∈ Γ_{T} (x̃) : (γu, γv) ∉ O\}|}{|Γ_{T} (x̃)|} = 0

uniformly on u, v.

In the case of the real hyperbolic space, Theorem 2 was proved in [Ma] using geometric methods.

2. Proof of Theorem 2

2.1. Cartan decomposition. Let G = K exp(𝔭) be the Cartan decomposition of G and A ⊂ exp(𝔭) a split Cartan subgroup of G, that is, a maximal connected abelian subgroup in exp(𝔭). We fix a system of positive roots Σ^{+} on 𝔞 = Lie(A), and let

A^{+} = {a ∈ A : α(log a) ≥ 0 for all α ∈ Σ^{+}}

denote the closed positive Weyl chamber in A. Then G = KA^{+}K, and a Haar measure on G can be given by

(2) ∫_{G} ψ(g)dg = ∫_{K} ∫_{A^{+}} ∫_{K} ψ(k_{1}ak_{2})ξ(log a)dν(k_{1})dadν(k_{2}), ψ ∈ C_{c}(G),

where da denotes the Lebesgue measure on A,

ξ(s) = ∏_{α∈Σ^{+}} sinh(α(s))^{m_{α}}, s ∈ 𝔞,

and m_{α} denotes the dimension of the root space for the root α ∈ Σ^{+}.

Let g̃ ∈ G be such that xg̃ = x̃. Then G = g̃^{−1}KA^{+}K, G_{T} (x̃) = g̃^{−1}KA_{T}^{+}K, and

(3) ∫_{G} ψ(g)dg = ∫_{K} ∫_{A^{+}} ∫_{K} ψ(g̃^{−1}k_{1}ak_{2})ξ(log a)dν(k_{1})dadν(k_{2}), ψ ∈ C_{c}(G).

In particular, it follows that

(4)  Vol(G_{T} (x̃)) = Vol(G_{T}) = ∫_{A_{T}^{+}} ξ(log a)da.

2.2. Reduction to maximal parabolics. Fix a system of simple roots

Π = {α_{1}, …, α_{r}} ⊂ Σ^{+}.

Here r = dim A is the ℝ-rank of G. It is well-known that the closed subgroups of G that contain P are in one-to-one correspondence with the subsets of Π (see [W, Sec. 1.2]). In particular, P_{i} = P_{{α_{i}}}, i = 1, …, r, are the maximal parabolic subgroups of G and

P = ⋂_{i=1}^{r} P_{i}.

We consider the projection maps

π_{i} : G/P × G/P → G/P_{i} × G/P_{i}, i = 1, …, r.

Let Δ and Δ_{i} denote the diagonals in G/P × G/P and G/P_{i} × G/P_{i} respectively. Then

Δ = ⋂_{i=1}^{r} π_{i}^{−1}(Δ_{i}).

Since

∏_{i=1}^{r} π_{i} : G/P × G/P → ∏_{i=1}^{r} G/P_{i} × G/P_{i}

is a continuous injective map from a compact space to a Hausdorff space, it is a homeomorphism onto its image. It follows that for any neighborhood O of Δ in G/P × G/P, there exist neighborhoods O_{i} of Δ_{i} in G/P_{i} × G/P_{i} such that

O ⊃ ⋂_{i=1}^{r} π_{i}^{−1}(O_{i}).

Then for every (u, v) ∈ G/P × G/P,

{g ∈ G : g · (u, v) ∉ O} ⊂ ⋃_{i=1}^{r}{g ∈ G : g · π_{i}(u, v) ∉ O_{i}}.

This inclusion shows that it suffices to prove Theorem 2 under the assumption that P is a maximal parabolic subgroup of G. We keep this assumption until the end of this section.

2.3. Dynamics on projective space. By a result from [T], there is an irreducible representation G → GL(V) such that the highest weight space is one-dimensional, and the stabilizer of this space is P. We consider the induced action of G on the projective space ℙ(V), and let w^{+} ∈ ℙ(V) be the direction of the highest weight space. The map g ↦ gw^{+} defines an embedding of G/P in ℙ(V). Note that if λ is the highest weight, the other weights of the representation are of the form λ−∑_{α∈Σ^{+}} n_{α}α for integers n_{α} ≥ 0. We denote by V^{<} the sum of all root spaces with weights other than λ. We fix a K-invariant scalar product on V, which gives rise to a metric d on ℙ(V), which is K-invariant. Put d̃(w_{1}, w_{2}) = d(g̃w_{1}, g̃w_{2}). Let V_{ε}^{<} be the open ε-neighborhood of V^{<} in ℙ(V) with respect to the metric d̃.

For w ∈ ℙ(V) and τ > 0, define

K_{τ}(w) = {k ∈ K : kw ∉ V_{τ}^{<}}.

Lemma 3. For every w ∈ G · w^{+},

lim_{τ→0^{+}} ν(K − K_{τ}(w)) = 0.

Proof. It follows from the Iwasawa decomposition that G·w^{+} = K ·w^{+}. Thus, without loss of generality, we may assume that w = w^{+}. By the continuity of the measure, it suffices to prove that

ν({k ∈ K : kw^{+} ∈ V^{<}}) = 0.

Suppose that this is false. For a subspace W of V, define

K_{W} = {k ∈ K : kw^{+} ∈ W }.

Let W be a minimal subspace of V^{<} such that ν(K_{W}) > 0. We claim that Stab_{K}(W) = K. If Stab_{K}(W) has infinite index in K, then there exist k_{i} ∈ K, i ≥ 1, such that k_{i}W ≠ k_{j}W for i ≠ j. Since all sets k_{i}K_{W} ⊂ K, i ≥ 1, have the same positive measure, it follows that for some i ≠ j, k_{i}K_{W} ∩ k_{j}K_{W} has positive measure. Then k_{j}^{−1}k_{i}K_{W} ∩ K_{W} has positive measure too, and for k ∈ k_{j}^{−1}k_{i}K_{W} ∩ K_{W},

kw^{+} ∈ k_{j}^{−1}k_{i}W ∩ W.

Since k_{j}^{−1}k_{i}W ∩ W is a proper subspace of W, this contradicts the choice of W. Thus, Stab_{K}(W) is a closed subgroup of finite index in K. Since K is connected, it follows that K = Stab_{K}(W). Then w^{+} ∈ K_{W}^{−1}W ⊂ V^{<}. This contradiction proves the lemma. □

We consider the sets

A_{T}^{η} = {a ∈ A_{T} : α(log a) ≥ η for all α ∈ Σ^{+}},
(5)  G_{T,ε}(u, v) = {g ∈ G_{T} (x̃) : d̃(gu, gv) > ε},
Ω_{T,τ}^{η}(u, v) = g̃^{−1}KA_{T}^{η} (K_{τ}(u) ∩ K_{τ}(v))

defined for T, η, τ, ε > 0 and u, v ∈ ℙ(V). Lemma 4. For every ε > 0 and τ > 0, there exists η > 0 such that for every T > 0 and u, v ∈ G · w^{+},

(6)  Ω_{T,τ}^{η}(u, v) ∩ G_{T,ε}(u, v) = ∅.

Proof. Note that an element a ∈ A_{T}^{η} acts by diagonal transformations on V with respect to some fixed basis, and the eigenvalue associated to the vector w^{+} is at least e^{η} times greater than the other eigenvalues. Therefore, for all w ∉ V_{τ}^{<} and sufficiently large η (depending only on τ and ε), we have d(aw, w^{+}) < ε/2 when a ∈ A_{T}^{η}. Thus, for

g̃^{−1}k_{1}ak_{2} ∈ Ω_{T,τ}^{η}(u, v) = g̃^{−1}KA_{T}^{η} (K_{τ}(u) ∩ K_{τ}(v)),

we have

d̃(g̃^{−1}k_{1}ak_{2}u, g̃^{−1}k_{1}ak_{2}v) = d(ak_{2}u, ak_{2}v) ≤ d(ak_{2}u, w^{+}) + d(ak_{2}v, w^{+}) < ε,

This proves the lemma. □

2.4. Completion of the proof. By (3),

(7)  Vol(Ω_{T,τ}^{η}(u, v)) = (∫_{A_{T}^{η}} ξ(log a)da) · ν(K_{τ}(u) ∩ K_{τ}(v)).

Let ε, δ ∈ (0, 1). Using Lemma 3, we choose τ > 0 such that

ν(K_{τ}(u) ∩ K_{τ}(v)) > 1 − δ.

Let η > 0 be as Lemma 4. By Lemma 9(a), for sufficiently large T,

∫_{A_{T}^{η}} ξ(a)da ≥ (1 − δ) ∫_{A_{T}^{+}} ξ(log a)da.

Thus, it follows from (4) and (7) that

Vol(Ω_{T,τ}^{η}(u, v)) ≥ (1 − δ)^{2}Vol(G_{T} (x̃)).

for sufficiently large T > 0. Therefore, by (6),

Vol(G_{T,ε}(u, v)) ≤ (1 − (1 − δ)^{2})Vol(G_{T} (x̃))

for all δ ∈ (0, 1) and sufficiently large T > 0. Since the sets

{(g_{1}P, g_{2}P) : d̃(g_{1}w^{+}, g_{2}w^{+}) < ε}, ε > 0,

form a base of the neighborhoods of the diagonal in G/P × G/P, this proves the first part of Theorem 2.

To prove the second part of Theorem 2, we choose a neighborhood P of e in G and a neighborhood Q of the diagonal in G/P × G/P such that

(8)  P^{−1}P ∩ Γ = {e},
(9)  P^{−1} · Q ⊂ O,
(10)  P · G_{T} (x̃) ⊂ G_{T +c}(x̃).

for fixed c > 0 and all T > 0. Here we use that Γ is discrete, the space G/P is compact, and the metric on the symmetric space is uniformly continuous. By (9), for every γ ∈ Γ such that γ · (u, v) ∉ O, we have Pγ · (u, v) ∩ Q = ∅. Thus, using (10), we deduce that

P · {γ ∈ Γ_{T} (x̃) : γ · (u, v) ∉ O} ⊂ {g ∈ G_{T +c}(x̃) : g · (u, v) ∉ Q}.

Then by (8), Pγ_{1} ∩ Pγ_{2} = ∅ for γ_{1}, γ_{2} ∈ Γ, γ_{1} ≠ γ_{2}, and

|\{γ ∈ Γ_{T} (x̃) : γ · (u, v) ∉ O\}| ≤ \frac{1}{Vol(P)}Vol(\{g ∈ G_{T +c}(x̃) : g · (u, v) ∉ Q\})
= o(Vol(G_{T +c}(x̃)))

as T → ∞. Now the second statement of Theorem 2 follows from Lemma 9(d) and (1).

3. Equidistribution on Γ\G

Recall that K is a maximal compact subgroups of G, and ν is the probability Haar measure on K. Denote by ϱ a right Haar measure on the minimal parabolic subgroup P. For a suitable normalization of ϱ, the Haar measure on G is given by

(11)  ∫_{G} ψ(g)dg = ∫_{K} ∫_{P} ψ(kp)dϱ(p)dν(k), ψ ∈ C_{c}(G).

We also define a measure μ on G by

(12)  ∫_{G} ψ(g)dμ(g) = ∫_{K} ∫_{P} ψ(kp^{−1})dϱ(p)dν(k), ψ ∈ C_{c}(G).

Note that μ is left K-invariant.

The first step in the proof of Theorem 1 is the following result. Proposition 5. For every Ψ ∈ C_{c}(Γ\G) and z ∈ Γ\G,

lim_{T →∞} \frac{1}{μ(G_{T})} ∫_{G_{T}} Ψ(zg)dμ(g) = \frac{1}{Vol(Γ\G)} ∫_{Γ\G} Ψ dg

where G_{T} = {g ∈ G : d(x, xg) < T }.

Proposition 5 is a consequence of the equidistribution of translates of K in Γ\G proved by Eskin and McMullen in [EM] (see also [S] for a more general result). They showed that for every strongly divergent sequence {g_{n}} ⊂ G,

(13)  lim_{n→∞} ∫_{K} Ψ(zkg_{n})dν(k) = \frac{1}{Vol(Γ\G)} ∫_{Γ\G} Ψ dg.

Recall that a sequence {g_{n}} ⊂ G is strongly divergent if the projection of {g_{n}} on every noncompact simple factor of G is divergent. Note that (13) was proved in [EM] under the condition that the lattice Γ is irreducible. Since the proof of (13) is based on mixing properties of the action of G on Γ\G, it is applicable to the case of a reducible lattice Γ provided that the sequence {g_{n}} is strongly divergent.

Denote by π_{i} : G → G_{i}, i = 1, …, s, the projections of G onto its simple factors. Let C_{i,j} ⊂ G_{i}, j ≥ 1, be an increasing sequence of compact subsets such that G_{i} = ∪_{j≥1}C_{i,j}. Define

(14)  G_{T,n} = G_{T} − ⋃_{1≤i≤s} π_{i}^{−1}(C_{i,n}).

Lemma 6. For every n ≥ 1, μ(G_{T,n}) ∼ μ(G_{T}) as T → ∞.

Proof. It suffices to show that for every i = 1, …, s and n ≥ 1,

μ(G_{T} ∩ π_{i}^{−1}(C_{i,n})) = o(μ(G_{T})) as T → ∞.

Fix i = 1, …, s and n ≥ 1. Note that G = DH, where D and H = ker(π_{i}) are normal connected semisimple Lie subgroups with finite centers, and D and H commute. We have π_{i}^{−1}(C_{i,n}) = D_{i,n}H for some compact set D_{i,n} ⊂ D. There is a constant δ > 0 such that

(15)  D_{i,n}H_{T −δ} ⊂ (D_{i,n}H)_{T} ⊂ D_{i,n}H_{T +δ} for all T > 0.

We define measures μ_{D} and μ_{H} for the groups D and H respectively as in (12). With appropriate normalization, μ = μ_{D} ⊗ μ_{H}. Thus, it follows from (15) that

(16)  μ(G_{T} ∩ π_{i}^{−1}(C_{i,n})) = μ((D_{i,n}H)_{T}) ≪ μ_{H}(H_{T +δ}).

Since G_{T} = KP_{T} and P_{T}^{−1} = P_{T}, using (11) and (12), we conclude that

(17)  μ(G_{T}) = ϱ(P_{T}^{−1}) = ϱ(P_{T}) = Vol(G_{T}).

Similarly, H = LQ_{T} where L is a maximal compact subgroup of H contained in K, and Q is a minimal parabolic subgroup of H. As in (17), we deduce that μ_{H}(H_{T}) = Vol_{H}(H_{T}). By (16), it is sufficient to show that

(18)  Vol_{H}(H_{T +δ}) = o(Vol(G_{T})) as T → ∞.

Note that with appropriate normalization the Haar measure on G is the product of Haar measures on D and H. Without loss of generality, Vol_{D}(D_{i,n}) > 0. Then by (15),

Vol_{H}(H_{T +δ}) ≪ Vol(D_{i,n}H_{T +δ}) ≤ Vol((D_{i,n}H)_{T +2δ}).

Let G_{T}^{η} be defined as in (24). Since the set D_{i,n} is compact, there exists η > 0 such that

(D_{i,n}H)_{T +2δ} ⊂ G_{T +2δ} − G_{T +2δ}^{η}.

Thus, (18) follows from Lemma 9(b). □

Proof of Proposition 5. The map K × A^{+} × K → G is a diffeomorphism on an open set of full measure. Since the measure μ is left K-invariant and smooth, for some σ ∈ C(A^{+} × K),

∫_{G} ψ(g)dμ(g) = ∫_{K} ∫_{A^{+}} ∫_{K} ψ(k_{1}ak_{2})σ(a, k_{2})dν(k_{1})dadν(k_{2}), ψ ∈ C_{c}(G).

Let G_{T,n} be defined as in (14), and it is K-bi-invariant (equivalently, all C_{i,j} are π_{i}(K)-bi-invariant). Then

G_{T,n} = KA_{T,n}^{+}K and μ(G_{T,n}) = ∫_{K} ∫_{A_{T,n}^{+}} σ(a, k_{2})dadν(k_{2}),

where A_{T,n}^{+} = G_{T,n} ∩ A^{+}.

Let ε > 0. By (13),

|∫_{K} Ψ(zk_{1}ak_{2})dν(k_{1}) − \frac{1}{Vol(Γ\G)} ∫_{Γ\G} Ψ dg| < ε

for a ∈ A_{T,n}^{+} and k_{2} ∈ K when n > n_{0}(ε). Thus, for n > n_{0}(ε),

|∫_{G_{T,n}} Ψ(zg)dμ(g) − \frac{μ(G_{T,n})}{Vol(Γ\G)} ∫_{Γ\G} Ψ dg| (19)

= |∫_{K} ∫_{A_{T,n}^{+}} ∫_{K} Ψ(zk_{1}ak_{2})dν(k_{1})σ(a, k_{2})dadν(k_{2})

− \frac{μ(G_{T,n})}{Vol(Γ\G)} ∫_{Γ\G} Ψ dg| ≤ ∫_{K} ∫_{A_{T,n}^{+}} |∫_{K} Ψ(zk_{1}ak_{2})dν(k_{1})

− \frac{1}{Vol(Γ\G)} ∫_{Γ\G} Ψ dg| σ(a, k_{2})dadν(k_{2}) < εμ(G_{T,n}).

By Lemma 6, for every n ≥ 1,

∫_{G_{T}} Ψ(zg)dμ(g) = ∫_{G_{T,n}} Ψ(zg)dμ(g) + o(μ(G_{T,n}))

as T → ∞. Thus, it follows from (19) that

lim sup_{T →∞} |\frac{1}{μ(G_{T})} ∫_{G_{T}} Ψ(zg)dμ(g) − \frac{1}{Vol(Γ\G)} ∫_{Γ\G} Ψ dg| < ε

for every ε > 0. This proves the proposition. □

4. Equidistribution on average

In this section we prove that Theorem 1 holds “on average”. In the case of hyperbolic spaces, the following proposition is a consequence of the work of Roblin [R]. Proposition 7. For every f ∈ C(G/P) and y ∈ G/P,

lim_{T →∞} \frac{1}{|Γ_{T} (x̃)|} ∑_{γ∈Γ_{T} (x̃)} ∫_{K} f(γky)dν(k) = ∫_{G/P} fdmx̃

where Γ_{T} (x̃) = {γ ∈ Γ : d(x, x̃γ) < T }.

Proof. There exists p̃ ∈ P such that x̃ = xp̃. Then K̃ = p̃^{−1}Kp̃, and it follows from (11) that

(20)  ∫_{G} ψ(g)dg = ∫K̃ ∫_{P} ψ(kp̃^{−1}p)dϱ(p)dν̃(k), ψ ∈ C_{c}(G).

Without loss of generality, f ≥ 0, and since G = KP, we may assume that y = eP. Let ε > 0, O_{ε} = {z ∈ X : d(x, z) < ε}, and ϕ_{ε} ∈ C_{c}(X) such that

ϕ_{ε} ≥ 0, supp(ϕ_{ε}) ⊂ O_{ε}, ∫_{P} ϕ_{ε}(xp^{−1})dϱ(p) = 1.

10 A. GORODNIK AND F. MAUCOURANT

Since X = x̃P and ϱ is right invariant, it follows that

(21)  ∫_{P} ϕ_{ε}(zp^{−1})dϱ(p) = 1 for every z ∈ X.

Let

ψ_{ε}(g) = f(gP)ϕ_{ε}(x̃g), g ∈ G.

Clearly, ψ_{ε} ∈ C_{c}(G) and

Ψ_{ε}(Γg) =^{def} ∑_{γ∈Γ} ψ_{ε}(γg) ∈ C_{c}(Γ\G).

By Proposition 5,

(22)  lim_{T →∞} \frac{1}{μ(G_{T})} ∑_{γ∈Γ} ∫_{G_{T}} ψ_{ε}(γg)dμ(g) = \frac{1}{Vol(Γ\G)} ∫_{Γ\G} Ψ_{ε}(Γg)dg

and by (20),

Vol(Γ\G) ∫_{Γ\G} Ψ_{ε}(Γg)dg = ∫_{G} ψ_{ε}(g)dg = ∫K̃ f(kP)dν̃(k) · ∫_{P} ϕ_{ε}(x̃p̃^{−1}p)dϱ(p)
= ∫_{G/P} fdmx̃ · ∫_{P} ϕ_{ε}(xp)dϱ(p).

Denote by δ the modular function of P. By (21),

|∫_{P} ϕ_{ε}(xp)dϱ(p) − 1| = |∫_{P} ϕ_{ε}(xp^{−1})(δ(p) − 1)dϱ(p)|
≤ max{|δ(p) − 1| : xp^{−1} ∈ O_{ε}}.

The sets {p ∈ P : xp^{−1} ∈ O_{ε}}, ε > 0, form a base of neighborhoods of P ∩ K in P. Since δ|_{P ∩K} = 1 and P ∩ K is compact,

max{|δ(p) − 1| : xp^{−1} ∈ O_{ε}} → 0 as ε → 0^{+}.

Thus, it follows from (22) that

(23) lim_{ε→0^{+}} lim_{T →∞} \frac{1}{μ(G_{T})} ∑_{γ∈Γ} ∫_{G_{T}} ψ_{ε}(γg)dμ(g) = ∫_{G/P} fdmx̃.

Since G_{T} = KP_{T},

∑_{γ∈Γ} ∫_{G_{T}} ψ_{ε}(γg)dμ(g)

=^{(12)} ∑_{γ∈Γ} ∫_{K×P_{T}^{−1}} ψ_{ε}(γkp^{−1})dν(k)dϱ(p)
= ∑_{γ∈Γ} ∫_{K} f(γkP) (∫_{P_{T}^{−1}} ϕ_{ε}(x̃γkp^{−1})dϱ(p)) dν(k).

PROXIMALITY AND EQUIDISTRIBUTION ON THE FURSTENBERG BOUNDARY 11

For γ ∈ Γ − Γ_{T +ε}(x̃), k ∈ K, and p ∈ P_{T}^{−1},

d(x, x̃γkp^{−1}) = d(xpk^{−1}, x̃γ) ≥ d(x, x̃γ) − d(x, xpk^{−1}) ≥ ε.

This implies that ∫_{P_{T}^{−1}} ϕ_{ε}(x̃γkp^{−1})dϱ(p) = 0 for γ ∈ Γ − Γ_{T +ε}(x̃). Thus,

∑_{γ∈Γ} ∫_{G_{T}} ψ_{ε}(γg)dμ(g)

= ∑_{γ∈Γ_{T +ε}(x̃)} ∫_{K} f(γkP) (∫_{P_{T}^{−1}} ϕ_{ε}(x̃γkp^{−1})dϱ(p)) dν(k)
≤ ∑_{γ∈Γ_{T +ε}(x̃)} ∫_{K} f(γkP) (∫_{P} ϕ_{ε}(x̃γkp^{−1})dϱ(p)) dν(k)
=^{(21)} ∑_{γ∈Γ_{T +ε}(x̃)} ∫_{K} f(γkP)dν(k).

Combining (23), (17), (1) and Lemma 9(c), we deduce that

lim inf_{T →∞} \frac{1}{|Γ_{T} (x̃)|} ∑_{γ∈Γ_{T} (x̃)} ∫_{K} f(γkP)dν(k) ≥ ∫_{G/P} fdmx̃.

On the other hand, for γ ∈ Γ_{T −ε}(x̃), k ∈ K, and p ∈ P such that d(x, x̃γkp^{−1}) < ε,

d(x, xp^{−1}) ≤ d(x, x̃γkp^{−1}) + d(xp^{−1}, x̃γkp^{−1}) < T.

This shows that for γ ∈ Γ_{T −ε}(x̃),

∫_{P_{T}^{−1}} ϕ_{ε}(x̃γkp^{−1})dϱ(p) = ∫_{P} ϕ_{ε}(x̃γkp^{−1})dϱ(p) =^{(21)} 1.

Hence,

∑_{γ∈Γ} ∫_{G_{T}} ψ_{ε}(γg)dμ(g)

≥ ∑_{γ∈Γ_{T −ε}(x̃)} ∫_{K} f(γkP) (∫_{P_{T}^{−1}} ϕ_{ε}(x̃γkp^{−1})dϱ(p)) dν(k)
= ∑_{γ∈Γ_{T −ε}(x̃)} ∫_{K} f(γkP)dν(k).

By (23), (17), (1), and Lemma 9(c),

lim sup_{T →∞} \frac{1}{|Γ_{T} (x̃)|} ∑_{γ∈Γ_{T} (x̃)} ∫_{K} f(γkP)dν(k) ≤ ∫_{G/P} fdmx̃.

This proves the proposition. □

5. Proof of Theorem 1

Now the proof can be completed using the argument from [Ma]. Let ε > 0. Since the space G/P × G/P is compact, there exists a neighborhood O of the diagonal in G/P × G/P such that for every (z_{1}, z_{2}) ∈ O, we have |f(z_{1}) − f(z_{2})| < ε. Then for every k ∈ K,

| ∑_{γ∈Γ_{T} (x̃)} f(γy) − ∑_{γ∈Γ_{T} (x̃)} f(γky)|

≤ ∑_{γ∈Γ_{T} (x̃):(γy,γky)∈O} |f(γy) − f(γky)| + ∑_{γ∈Γ_{T} (x̃):(γy,γky)∉O} |f(γy) − f(γky)|
≤ ε|Γ_{T} (x̃)| + 2 sup |f| · |{γ ∈ Γ_{T} (x̃) : (γy, γky) ∉ O}|.

Thus, it follows from Theorem 2 that

lim_{T →∞} \frac{1}{|Γ_{T} (x̃)|} | ∑_{γ∈Γ_{T} (x̃)} f(γy) − ∑_{γ∈Γ_{T} (x̃)} f(γky)| = 0

for all k ∈ K. Hence, by the dominated convergence theorem,

lim_{T →∞} |\frac{1}{|Γ_{T} (x̃)|} ∑_{γ∈Γ_{T} (x̃)} f(γy) − \frac{1}{|Γ_{T} (x̃)|} ∑_{γ∈Γ_{T} (x̃)} ∫_{K} f(γky)dν(k)| = 0.

Finally, Theorem 1 follows from Proposition 7.

6. Appendix: volume estimates

In this section, we give proofs of volume estimates, which are used in Theorems 1 and 2. There are other ways to establish these volume estimates. For example, one can use the exact asymptotics of the volume of Riemannian balls from [Kn] (see also [GO]). We present a straightforward proof that does not use asymptotics.

Let 𝔞 be the Lie algebra of the Cartan subgroup A and 𝔞^{+} the positive Weyl chamber with respect to the root system Σ^{+}. The Riemannian metric defines a scalar product on 𝔞 and, by duality, on the dual space of 𝔞. For α ∈ Σ^{+}, we denote by m_{α} the dimension of the corresponding root space and put ρ = \frac{1}{2} ∑_{β∈Σ^{+}} m_{β}β. Lemma 8. The maximum of ρ on {a ∈ 𝔞 : ∥a∥ ≤ 1} is achieved at a unique point in the interior of 𝔞^{+}.

Proof. Since the set {a ∈ 𝔞 : ∥a∥ = 1} is strictly convex, it is clear that the point of maximum is unique. It is sufficient to show that (ρ, α) > 0 for every α ∈ Σ^{+}. Denote by σ_{α} the reflection with respect to the hyperplane {α = 0}. The map σ_{α} permutes the elements of the set Σ^{+} − {α, 2α} and σ_{α}(α) = −α. Since m_{σ_{α}(β)} = m_{β}, we have

σ_{α}(ρ) = ρ − 2m_{α}α − 4m_{2α}α.

PROXIMALITY AND EQUIDISTRIBUTION ON THE FURSTENBERG BOUNDARY 13

Thus,

(ρ, α) = (σ_{α}(ρ), σ_{α}(α)) = 2m_{α}(α, α) + 4m_{2α}(α, α) − (ρ, α)

and (ρ, α) = (m_{α} + 2m_{2α})(α, α) is positive. □

For T, η > 0, define

A_{T}^{η} = {a ∈ A_{T} : α(log a) ≥ η for all α ∈ Σ^{+}}
(24)  = {a ∈ A : ∥ log a∥ < T, α(log a) ≥ η for all α ∈ Σ^{+}},
G_{T}^{η} = KA_{T}^{η} K.

Lemma 9. For every η > 0,

(a)

∫_{A_{T}^{η}} ξ(log a)da ∼_{T →∞} ∫_{A_{T}^{+}} ξ(log a)da,

(b)

Vol(G_{T}^{η}) ∼_{T →∞} Vol(G_{T}),

(c)

lim inf_{ε→0^{+}} (lim sup_{T →∞} \frac{Vol(G_{T +ε})}{Vol(G_{T})}) = 1,

(d)

Vol(G_{T +η}) ≪ Vol(G_{T}).

Proof. We have

(25)  ∫_{𝔞_{T}^{+}} ξ(a)da = 2^{−|Σ^{+}|} ∑_{i∈I} ∫_{𝔞_{T}^{+}} e^{λ_{i}(a)}da

where λ_{i}’s the characters of the form 2ρ − ∑_{α∈Σ^{+}} n_{α}α for some n_{α} ≥ 0. Let

δ = max{2ρ(a) : a ∈ 𝔞_{1}^{+}},
δ_{i} = max{λ_{i}(a) : a ∈ 𝔞_{1}^{+}}, i ∈ I,
δ_{α} = max{2ρ(a) : a ∈ 𝔞_{1}^{+}, α(a) = 0}, α ∈ Σ^{+}.

It follows from Lemma 8 that for λ_{i} ≠ 2ρ and α ∈ Σ^{+}, δ > max{δ_{i}, δ_{α}}. Thus,

(26)  ∫_{𝔞_{T}^{+}} e^{λ_{i}(a)}da ≤ Vol(𝔞_{T}^{+})e^{δ_{i}T} ≪ T^{r}e^{δ_{i}T}

where r = dim 𝔞. Let ε > 0 be such that

δ − ε > max{δ_{i}, δ_{α} : i ∈ I, α ∈ Σ^{+}}.

Then

(27) ∫_{𝔞_{T}^{+}} e^{2ρ(a)}da = T^{r} ∫_{𝔞_{1}^{+}} e^{2T ρ(a)}da
≥ T^{r}e^{(δ−ε)T} Vol({a ∈ 𝔞_{1}^{+} : 2ρ(a) ≥ δ − ε}) ≫ T^{r}e^{(δ−ε)T} .

Combining (25), (26), and (27), we deduce that

(28)  ∫_{𝔞_{T}^{+}} ξ(a)da ≫ T^{r}e^{(δ−ε)T} .

On the other hand, for α ∈ Σ^{+},

∫_{𝔞_{T}^{+}∩{α<η}} ξ(a)da ≤ ∫_{𝔞_{T}^{+}∩{α<η}} e^{2ρ(a)}da ≪ ∫_{𝔞_{T}^{+}∩{α=0}} e^{2ρ(a)}da
= T^{r−1} ∫_{𝔞_{1}^{+}∩{α=0}} e^{2T ρ(a)}da ≪ T^{r−1}e^{δ_{α}T} = o(e^{(δ−ε)T}).

Since

𝔞_{T}^{+} − 𝔞_{T}^{η} ⊂ ⋃_{α∈Σ^{+}} 𝔞_{T}^{+} ∩ {α < η}.

This proves part (a) of the lemma. Part (b) follows from (2).

To prove part (c), we note that

Vol(G_{T +ε}) = ∫_{𝔞_{T +ε}^{+}} ξ(a)da = (T + ε)^{r} ∫_{𝔞_{1}^{+}} ξ((T + ε)a)da

It is easy to check that there exist b > 0 such that sinh(t + ε) ≤ e^{ε} sinh(t) + b for every ε ∈ (0, 1) and t ≥ 0. Thus, for a ∈ 𝔞_{1}^{+} and sufficiently small ε > 0,

ξ((T + ε)a) ≤ ∏_{α∈Σ^{+}}(a_{ε} sinh(α(T a)) + b)^{m_{α}} ≤ d_{ε}ξ(T a) + C ∑_{i∈I} e^{λ_{i}(a)}

where d_{ε} → 1 as ε → 0^{+}, C > 0, and λ_{i}’s are characters such that 2ρ − λ_{i} < 0 in the interior of 𝔞^{+}. Thus, it follows from (26) that

∫_{𝔞_{T}^{+}} ξ((T + ε)a)da ≤ d_{ε} ∫_{𝔞_{T}^{+}} ξ(T a)da + o(e^{(δ−ε)T}).

Using (4) and (28), we deduce that

lim sup_{T →∞} \frac{Vol(G_{T +ε})}{Vol(G_{T})} ≤ d_{ε},

and part (c) of the lemma follows. The last part of lemma can be proved similarly. □

7. Acknowledgements

The main ideas of this paper were developed during the workshop “Ergodic properties of geometric group actions” in Summer 2003. The authors would like to express deep appreciation to the organizers of this workshop and to the Max Planck Institute of Mathematics for its support. We also would like to thank to R. Spatzier for raising the problem solved in this paper during the workshop and to Y. Guivarc’h for bringing the paper of Arnold and Krylov to out attention.

References

[AK] V. I. Arnold and A. L. Krylov, Uniform distribution of points on a sphere and certain ergodic properties of solutions of linear ordinary differential equations in a complex domain. Dokl. Akad. Nauk SSSR 148, 9–12, 1963.

[DRS] W. Duke,Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties. Duke Math. J. 71 (1993), no. 1, 143–179.

[EM] A. Eskin and C. McMullen, Mixing, counting and equidistribution on Lie groups. Duke Math. J. 71 (1993), no. 1, 181–209.

[F] H. Furstenberg, Boundary theory and stochastic processes on homogeneous spaces. Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI), 193–229. Amer. Math. Soc., Providence, R.I., 1973.

[G] A. Gorodnik, Lattice action on the boundary of SL(n, ℝ). Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1817–1837.

[GO] A. Gorodnik and H. Oh, Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary. Preprint.

[Ka] D. A. Každan, Uniform distribution on a plane, Trudy Moskov. Mat. Obšč. 14, 299–305, 1965.

[Kn] G. Knieper, On the asymptotic geometry of nonpositively curved manifolds. Geom. Funct. Anal. 7 (1997), no. 4, 755–782.

[L] F. Ledrappier, Distribution des orbites des réseaux sur le plan réel. C.R. Acad. Sci. Paris Sr. I Math. 329, no. 1, 61–64, 1999.

[Ma] F. Maucourant, Approximation diophantienne, dynamique des chambres de Weyl et répartitions d’orbites de réseaux. PhD Thesis, Université de Lille, 2002.

[Mo] G. D. Mostow, Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies, No. 78. Princeton University Press, 1973.

[N] A. Nogueira, Orbit distribution on ℝ^{2} under the natural action of SL(2, ℤ). Indag. Math. (N.S.) 13 (2002), no. 1, 103–124.

[R] T. Roblin, Ergodicité et équidistribution en courbure négative. Mémoires de la SMF 95 (2003).

[S] N. Shah, Limit distribution of expanding translates of certain orbits on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 2, 105–125.

[T] J. Tits, Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque. J. Reine Angew. Math. 247 (1971), 196–220.

[W] G. Warner, Harmonic analysis on semi-simple Lie groups. I. Die Grundlehren der mathematischen Wissenschaften, Band 188. Springer-Verlag, New York-Heidelberg, 1972.

Mathematics department, University of Michigan, Ann Arbor, MI 48109 USA

  E-mail address: gorodnik@umich.edu

École normale supérieure de Lyon, Unité de Mathématiques Pures et Appliquées, UMR CNRS 5669, 46, allée d’Italie, 69364 Lyon Cedex 07 France

  E-mail address: Francois.MAUCOURANT@umpa.ens-lyon.fr