Entropy rigidity of symmetric spaces without focal points

ENTROPY RIGIDITY OF SYMMETRIC SPACES WITHOUT FOCAL POINTS

arXiv:1205.1070v3 [math.DG] 16 Jan 2013

FRANC ¸ OIS LEDRAPPIER AND LIN SHU

To Werner Ballmann for his 61st birthday Abstract. We characterize symmetric spaces without focal points by the equality case of general inequalities between geometric quantities.

1. Introduction f, ge) → (M, g) its uniLet (M, g) be a closed connected Riemannian manifold, and π : (M versal cover endowed with the lifted Riemannian metric. We denote p(t, x, y), t ∈ R+ , x, y ∈ f, f the heat kernel on M f, the fundamental solution of the heat equation ∂u = ∆u on M M ∂t f. Since we have a compact quotient, all the where ∆ = Div ∇ is the Laplacian on M f: following limits exist as t → ∞ and are independent of x ∈ M R |∇f |2 1 R λ0 = inf = lim − ln p(t, x, x) 2 t t |f | f) f ∈Cc2 (M Z 1 ℓ = lim d(x, y)p(t, x, y) dVol(y) t t Z 1 p(t, x, y) ln p(t, x, y) dVol(y) h = lim − t t 1 v = lim ln VolBM f(x, t), t t f where BM f(x, t) is the ball of radius t centered at x in M and Vol is the Riemannian volume f. on M

All these numbers are nonnegative. Recall λ0 is the bottom of the spectrum of the Laplacian, ℓ the linear drift, h the stochastic entropy and v the volume entropy. There is the following relation: (a)

(1.1)

(b)

(c)

4λ0 ≤ h ≤ ℓv ≤ v 2 .

2000 Mathematics Subject Classification. 53C24, 53C20. Key words and phrases. volume entropy, rank one manifolds. 1

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See [L1] for (a), [Gu] for (b). Inequality (c) is shown in [L4] as a corollary of (b) and (1.2): (1.2)

ℓ2 ≤ h.

In this paper, we are interested in the characterization of locally symmetric property of M by the equality case of the above inequalities for manifolds without focal points. Recall that a Riemannian manifold M is said to have no focal points if for any imbedded open geodesic segment γ : (−a, a) 7→ M (where 0 < a ≤ ∞), the restriction of the exponential map on the normal bundle of γ is everywhere nonsingular. Hence any manifold of nonpositive curvature has no focal points. The reverse is not true since there exist manifolds without focal points but with sectional curvatures of both signs ([Gul]). If M is a locally symmetric space without focal points, it must have nonpositive curvature ([He, Theorem 3.1]). Note that for symmetric spaces of nonpositive curvature, 4λ0 = v 2 ([Kar], cf. [BCG1, Appendice C]) and hence all five numbers 4λ0 , ℓ2 , h, ℓv, v 2 coincide by (1.1), (1.2) above and are positive f, g) is (Rn , Eucl.). Our result is a partial converse: unless (M Theorem 1.1. Let (M, g) be a compact connected Riemannian manifold without focal points. With the above notation, all the following equalities are equivalent to the locally symmetric property of M : i) 4λ0 = v 2 ; ii) h = ℓ2 or v 2 ; iii) ℓ = v. As recalled in [L4], Theorem 1.1 is known in negative curvature case and follows from [K1], [BFL], [BCG2], [FL] and [L1]. The other possible converses are delicate: even for negatively curved manifolds, in dimension greater than two, it is not known whether h = ℓv holds only for locally symmetric spaces. This is equivalent to a conjecture of Sullivan (see [L3] for a discussion). Sullivan’s conjecture holds for surfaces of negative curvature ([L1], [Ka]). In negative curvature case, 4λ0 = h (and hence 4λ0 = ℓv) implies M is locally symmetric, which follows from [L2], [BFL], [BCG2], and [FL]. In the no focal points case, whether 4λ0 = h holds only for locally symmetric spaces may depend on a further f (cf. [AS] for this notion). However, it would follow study on the Martin boundary of M 2 from the hypothetical 4λ0 ≤ ℓ by ii) of Theorem 1.1.

We assume henceforth that (M, g) has no focal points. Given a geodesic γ in M , Jacobi fields along γ are vector fields t 7→ J(t) ∈ Tγ(t) M which describe infinitesimal variation of geodesics around γ. The rank of the geodesic γ is the dimension of the space of Jacobi fields such that t 7→ kJ(t)k is a constant function on R. The rank of a geodesic γ is at least one because of the trivial t 7→ γ(t) ˙ which describes the variation by sliding the geodesic along itself. The rank of the manifold M is the smallest rank of geodesics in M . Using rank rigidity theorem for manifolds with no focal points from [W], we reduce in section 2 the proof of Theorem 1.1 to proving that if (M, g) is rank one, the equality ℓ2 = h implies that f, ge) is a symmetric space. For this, we show in section 4 that the equality ℓ2 = h implies (M f, ge) is asymptotically harmonic (see section 2.4 for its definition). This uses the that (M

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solvability of the Dirichlet problem at infinity and the structure of harmonic measures of the stable foliation, which will be presented in section 3. Finally, it was recently observed by A. Zimmer ([Z1]) that asymptotically harmonic universal covers of rank one manifolds are indeed symmetric spaces.

2. Generalities and reduction of Theorem 1.1 We assume in the following that (M, g) is a compact connected Riemannian manifold f be the universal cover of M with covering group Γ = π1 (M ). without focal points. Let M

2.1. Geometric boundary. The notion of geometric boundary was introduced by Eberlein and O’Neill [EO] for nonpositive curvature case (see [B3]). Consider the geodesics f. Two geodesic rays γ1 and γ2 of M f are said to be asymptotic (or equivalent) if on M f(∞), is supt≥0 d(γ1 (t), γ2 (t)) < +∞. The set of equivalence classes [γ], denoted by M f. We denote by M c the union M f∪M f(∞). called the geometric boundary of M f, for any geodesic ray γ of M f, there exists a unique geodesic starting at x Given x ∈ M f× M f(∞), there is which is asymptotic to γ ([O, Proposition 3]). Hence for any (x, ξ) ∈ M f the a unique unit speed geodesic γx,ξ satisfying γx,ξ (0) = x and [γx,ξ ] = ξ. Denote by Sx M −1 f. The mapping πx : M f(∞) 7→ Sx M f sending ξ to γ˙ x,ξ (0) is unit tangent space at x to M a bijection between these two sets. f, p ∈ M f, the angle ∠p (v, w) is the unique number 0 ≤ θ ≤ π such that For v, w ∈ Sp M c other than p, let γp,q denote the unique unit speed geodesic hv, wi = cos θ. For q ∈ M f and 0 < ǫ < π. The set starting at p pointing at q. Given v ∈ Sp M c : ∠p (v, γ˙ p,q (0)) < ǫ} C(v, ǫ) := {q ∈ M

is called the cone of vertex p, axis v, and angle ǫ (cf. [EO]). It was shown in [Go] that c so that for any x ∈ M f, the mapping πx is a there exists a canonical topology on M f and M f(∞). The topology is called the “cone” topology in homeomorphism between Sx M f(∞), the truncated cones the sense for ξ ∈ M   c\B(p, r) C(v, ǫ, r) = C(v, ǫ) ∩ M containing ξ, where B(p, r) is the closed ball of radius r about p, form a local basis at ξ.

f with M f×M f(∞) by (x, v) 7→ (x, πx v). The action of Γ on M can We will identify S M f(∞). Hence the quotient SM is identified with the quotient be continuously extended to M f×M f(∞) under the diagonal action of Γ. of M

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f, ge) and let N (γ) be the normal 2.2. Stable Jacobi tensor. Let γ be a geodesic in (M bundle of γ: ⊥ N (γ) := ∪t∈R Nt (γ), where Nt (γ) = (γ(t)) ˙ = {X ∈ Tγ(t) M : hX, γ(t)i ˙ = 0}.

A (1,1) tensor along γ is a family V = {V (t), t ∈ R}, where V (t) is an endomorphism of Nt (γ) such that for any family Yt of parallel vectors along γ, the covariant derivative D V (t)Yt exists. V ′ (t)Yt := dt We endow N (γ) with Fermi orthonormal coordinates given by a parallel frame field along γ. A (1,1) tensor along γ is parallel if V ′ (t) = 0 for all t. It is then given by a constant matrix in Fermi coordinates. The curvature tensor R induces a symmetric (1,1) tensor along γ by R(t)X = R(X, γ(t)) ˙ γ(t). ˙ A (1,1) tensor V (t) along γ is called a Jacobi tensor if ′′ it satisfies V + RV = 0. If V (t) is a Jacobi tensor along γ, then J(t) := V (t)Yt is a Jacobi field for any parallel field Yt . f, let γv denote the unique geodesic starting from x with speed v. For each v = (x, v) ∈ S M A Jacobi tensor Vv defined for each γv is called continuous if the initial values Vv (0), Vv′ (0) f× TM f : w ⊥ v} are continuous as (1, 1) tensors of the vector bundle B := {(v, w) ∈ S M f. over S M f, denote by Av be the Jacobi tensor along γv with initial Now for v = (x, v) ∈ S M condition Av (0) = 0 and A′v (0) = I (I is the identity). For each s > 0, let Sv,s be the Jacobi tensor with the boundary conditions Sv,s (0) = I and Sv,s (s) = 0. It can be shown (cf. [EOs]) that the limit lims→+∞ Sv,s =: Sv exists and is given by Z +∞ Sv (t) = Av (A∗v Av )−1 (u) du, t

A∗v

is the transposed form of Av . The tensor Sv is called the stable tensor along where ′ (0) to S ′ (0) ([E, the geodesic γv . As a consequence of the uniform convergence of Sv,s v ′ (0) (with respect to v) that the stable tensor Proposition 5]), one has by continuity of Sv,s Sv is continuous with respect to v ([EOs, Proposition 4]).

f, the vectors (Y, Sv (0)Y ) describe variations of asymptotic For each v = (x, v) ∈ S M f they generate corresponds to T Wvs , where Wvs , geodesics and the subspace Evs ⊂ Tv T M f × πx (v) in the set of initial vectors of geodesics asymptotic to γv , is identified with M f f f f M × M (∞). Recall that SM is identified with the quotient of M × M (∞) under the s diagonal action of Γ. Clearly, for ϕ ∈ Γ, ϕ(Wvs ) = WDϕv so that the collection of Wvs define a foliation W s on SM , the so-called stable foliation of SM . The leaves of the stable f, they are naturally endowed with the Riemannian metric foliation W s are quotient of M induced from ge.

Similarly, by reversing the time in the construction of stable tensor, one obtains the corresponding unstable tensor and hence the unstable subspaces and the unstable foliation.

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f as a reference point. For each ξ ∈ M f(∞), define 2.3. Busemann functions. Fix x0 ∈ M a Busemann function at ξ (cf. [E]) as follows. Let γx0 ,ξ be the unique unit speed geodesic starting at x0 which is asymptotic to ξ. For each s ≥ 0, define the function f. bξ,s (x) := d(x, γx ,ξ (s)) − s, ∀x ∈ M 0

We have by triangle inequality that bξ,s (x) are decreasing with s and bounded absolutely from below by −d(x0 , x). So the function f, bξ (x) := lim bξ,s (x), ∀x ∈ M s→∞

is well defined and is called the Busemann function at ξ. It was shown in [E] that the function x 7→ bξ (x) is of class C 2 .

f×M f(∞), we consider bϕξ (ϕx). The geodesics γx ,ϕξ and For each ϕ ∈ Γ, (x, ξ) ∈ M 0 γϕx0 ,ϕξ are asymptotic. Hence the Busemann functions at ϕξ using different reference points x0 and ϕx0 only differ by a constant depending on x0 , ϕ, ξ ([E, Proposition 3]), which is in fact given by bξ (ϕx0 ). So we have bϕξ (ϕx) := = = =

lim (d(ϕx, γx0 ,ϕξ (s)) − s)

s→+∞

lim (d(ϕx, γϕx0 ,ϕξ (s)) − s) + bξ (ϕx0 )

s→+∞

lim (d(x, γx0 ,ξ (s)) − s) + bξ (ϕx0 )

s→+∞

bξ (x) + bξ (ϕx0 ).

It follows that the function ∆x bξ satisfies ∆ϕx bϕξ = ∆x bξ and therefore defines a function f×M f(∞)) = SM , which is called the Laplacian of the Busemann function. B on Γ\(M

f, let ξ = [γv ] and let bv := bx,ξ , where bx,ξ is the Busemann function For v = (x, v) ∈ S M at ξ using x as a reference point. It is true ([E]) that ∇w (∇bv ) = −Sv′ (0)(w).

Since the stable tensor Sv is continuous with respect to v, we have ∆x bv = −TrSv′ (0) also depends continuously on v. Note that ∆x bv = ∆x bξ , we have ∆x bξ depends continuously on ξ. Consequently the function B is continuous on SM .

f, ge) has no focal points. By 2.4. Proof of Theorem 1.1. We continue assuming that (M f, ge) is of the form the Rank Rigidity Theorem (see [W]), (M f0 × M f1 × · · · × M fj × M fj+1 × · · · × M fk , ge)1, (M

where ge is the product metric ge2 = (e g0 )2 + (e g1 )2 + · · · + (e gj )2 + (e gj+1 )2 + · · · + (e gk )2 , f0 , ge0 ) is Euclidean, (M fi , gei ) is an irreducible symmetric space of rank at least two for (M fi , gei ), i = j +1, · · · , k, are i = 1, · · · , j and a rank one manifold for i = j +1, · · · , k. If the (M 1With a clear convention for the cases when Dim M f0 = 0, j = 0 or k = j.

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f, ge) is a symmetric space. Moreover in that case, all symmetric spaces of rank one, then (M all inequalities in (1.1) are equalities: this is the case for irreducible symmetric spaces (all numbers are 0 for Euclidean space; for the other spaces, note that locally a symmetric space without focal points must have nonpositive curvature ([He, Theorem 3.1]) and 4λ0 and v 2 are classically known to coincide for locally symmetric space with nonpositive curvature (cf. [BCG1, Appendice C])) and we have: X X fi ), v 2 (M f) = fi ). f) = 4λ0 (M 4λ0 (M v 2 (M i

i

fi in the To prove Theorem 1.1, it suffices to prove that if = h (or 4λ0 = h), all M decomposition are symmetric spaces. This is already true for i = 0, 1, · · · , j. It remains to fi , gei ) are symmetric spaces for i = j + 1, · · · , k. Note that each one of the show that (M f spaces (Mi , gei ) admits a cocompact discrete group of isometries (see [Kn1, Theorem 3.3] using the corresponding theorems from [W]). This shows that the linear drifts ℓi and the fi , gei ). Moreover, we clearly have stochastic entropies hi exist for each one of the spaces (M X X ℓ2 = ℓ2i , h = hi . ℓ2

i

Therefore Theorem 1.1 follows from

Theorem 2.1. Assume (M, g) is a closed connected rank one manifold without focal points f, ge) is a symmetric space. and that ℓ2 = h. Then (M

Let (M, g) be a closed connected Riemannian manifold without focal points as before. f, ge) is said to be asymptotically harmonic if B, the Laplacian of the Its universal cover (M Busemann function, is constant on SM . In that case, we have by [Z1, Theorem 1.2] that (M, g) is either flat or the geodesic flow on SM is Anosov. The latter case, as was observed by Knieper [Kn2, Theorem 3.6], actually implies M is a rank one locally symmetric space. f being asymptotically harmonic. If the geodesic flow (Indeed, let M be as above with M on SM is Anosov, then it is true by P. Foulon and F. Labourie [FL] that the stable and unstable distribution E s and E u of the geodesic flow are C ∞ . Hence the result of Y. Benoist, P. Foulon and F. Labourie [BFL] applies and gives that the geodesic flow of (M, g) is smoothly conjugate to the geodesic flow of a locally symmetric space (M0 , g0 ) of negative curvature. Note that in the no focal points case, the volume entropy and the topological entropy of the geodesic flow coincide [FM]. Thus one can use G. Besson, G. Courtois and S. Gallot’s rigidity theorem ([BCG2]) to conclude that the two spaces (M, g) and (M0 , g0 ) are isometric.) In summary, we have Proposition 2.2. ([Z1, Theorem 1.1]) Assume (M, g) is a closed connected rank one f, ge) is asymptotically harmonic. Then (M f, ge) is manifold without focal points such that (M a symmetric space. Therefore, Theorem 2.1 directly follows from Proposition 2.2 and

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Proposition 2.3. Assume (M, g) is a closed connected rank one manifold without focal f, ge) is asymptotically harmonic. points and that ℓ2 = h. Then (M 3. Harmonic measure for the stable foliation We consider the stable foliation W := W s of subsection 2.2. Recall that the leaves are endowed with a natural Riemannian metric. We write ∆W for the associated Laplace operator on functions which are of class C 2 along the leaves of W. A probability measure m on SM is called harmonic if it satisfies, for any C 2 function f , Z ∆W f dm = 0. SM

Our main result in this section is:

Theorem 3.1. Let (M, g) be a closed connected rank one manifold without focal points, W the stable foliation on SM endowed with the natural metric as above. Then, there is only one harmonic probability measure m and the support of m is the whole space SM . f× M f(∞) locally A main step in the proof of Theorem 3.1 is to identify the lift of m in M f f as dx × dmx , where mx is the hitting probability at M (∞) of the Brownian motion on M f. We adopt Ballmann’s starting at x and dx is proportional to the Riemannian volume on M approach ([B2, B3]) to use Lyons-Sullivan’s procedure ([LS]) of discretizing the Brownian motion (starting at x) to a random walk on Γ and show mx is indeed the unique stationary measure of the corresponding random process on Γ. The argument involves random walk on Γ, the solvability of the Dirichlet problem at infinity and divergence properties of geodesics for manifolds without focal points and is divided into seven steps for clarity.

f. The discretization procedure of 3.1. Discretization of Brownian motion. Fix x0 ∈ M f a probability measure Lyons and Sullivan ([LS]) associates to the Brownian motion on M f ν on Γ with ν(ϕ) > 0 for all ϕ ∈ Γ such that any bounded harmonic function h on M satisfies X h(x0 ) = h(ϕx0 )ν(ϕ). ϕ∈Γ

Consider the random walk on Γ defined by ν with transition probability of ν(ϕ−1 ϕ) e from ϕ ∈ Γ to ϕ e ∈ Γ. For given ϕ1 , · · · , ϕk , the probability Q that a sequence {ϕn } begins with ϕ1 , · · · , ϕk is defined to be −1 ν(ϕ1 )ν(ϕ−1 1 ϕ2 ) · · · ν(ϕk−1 ϕk ).

f The random walk on Γ generated by ν is a good approximation of Brownian motion on M starting at x0 in the following sense:

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Proposition 3.2. ([LS, Theorem 6], cf. [B2, Sec. 4]) Let W be the space of all continuous f and Ω the space of all sequences of heads and tails. There is a paths c : (0, +∞) → M probability measure P on W × Ω with the following properties: i) The natural projection from W × Ω to W maps P to P, the probability measure on W associated to the Brownian motion starting at x0 . ii) There is a map W ×Ω → ΓN , (c, ω) → {ϕn (c, ω)}, which maps P onto the probability measure Q. iii) There is an increasing sequence of stopping times Tn on W × Ω and a positive constant δ < 1 such that   P max d(cω (t), ϕn (c, ω)) > k ≤ δk , ∀k > 0. Tn θα( βn) 2 √2 > 2ǫ n, 1

if we choose ǫ < 2−1 θα( 12 β) 2 . This, together with (3.12), contradicts (3.10).



3.7. Proof of Theorem 3.1. Proof. Let m be a W harmonic probability measure on SM . Then, there is a unique f which coincides with m locally. Seen as a measure on Γ-invariant measure m e on S M f×M f(∞), we claim that m M e is given, for any continuous f with compact support, by: ! Z Z Z (3.13) f (x, ξ) dm(x, e ξ) = f (x, ξ) dmx (ξ) dx, f M

f(∞) M

f(∞) starting at x and dx is where mx is the hitting measure of Brownian motion at M f. proportional to the Riemannian volume on M

f(∞) with (3.13) holds Firstly, there is a family of probability measures x 7→ mx on M R f(∞), x 7→ g(ξ) dmx (ξ) is a harmonic function on M f. such that, for all g continuous on M f This follows from [Ga]. On the one hand, the measure m e projects on M as a Γ-invariant R f is proportional to Volume measure satisfying ∆f dm = 0. The projection of m e on M and formula (3.13) is the desintegration formula. On the other hand, if one projects m e f f first on M (∞), there is a probability measure m on M (∞) such that Z  Z Z f (x, ξ) dm(x, e ξ) = f (x, ξ) dmξ (dx) dm(ξ). f(∞) M

f M

f; therefore, for m-a.e. ξ, For m-a.e. ξ, the measure mξ is a harmonic measure on M there is a positive R harmonic function kξ (x) such that mξ = kξ (x)Vol . Comparing the two expressions for f dm, e we see that, up to a normalizing constant, the measure mx is given by mx (dξ) = kξ (x)m(dξ). R We normalize m in such a way that kξ (x0 ) = 1 for almost all ξ. Then, x 7→ M f(∞) g(ξ) dmx (ξ) is indeed a harmonic function. f, let ν be the corresponding Lyons-Sullivan measure on Γ. Since Next, for any x0 ∈ M f(∞), x 7→ mx (g) is a harmonic function and that mϕx = ϕ∗ mx for all g continuous on M 0

0

ENTROPY RIGIDITY OF SYMMETRIC SPACES WITHOUT FOCAL POINTS

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for ϕ ∈ Γ, it follows that the measure mx0 is a stationary measure for ν, i.e. X mx 0 = ϕ∗ mx0 ν(ϕ). ϕ∈Γ

So we conclude from Theorem 3.3 and Theorem 3.12 that mx0 is the hitting probability f(∞) of the Brownian motion starting at x0 . Since x0 was arbitrary in the above at M reasoning, we have the desired expression of the lift of m as in (3.13).

Finally, we have by the solvability of Dirichlet problem (Theorem 3.8) that each mx f(∞). It follows that the unique W harmonic measure m is fully is fully supported on M supported on SM .  4. A linear drift characterization of local symmetry

For Theorem 2.1, it remains to show Proposition 2.3. Consider the action of Γ on c=M f∪M f(∞). Let XM be the quotient of the space M f×M c by the diagonal action of M c c f Γ. To each ξ ∈ M is associated the projection of Wξ of M × {ξ} in XM . As a subgroup of f and the space W cξ is homeomorphic Γ, the stabilizer Γξ of the point ξ acts discretely on M f by Γξ . Put on each W cξ the smooth structure and the metric inherited to the quotient of M f. The manifold W cξ and its metric vary continuously on XM . The collection of from M cξ , ξ ∈ M c form a continuous lamination W c of XM with leaves which are manifolds all W c W f. Denote by ∆ the laminated Laplace operator acting on functions locally modeled on M which are smooth along the leaves of the lamination. A Borel measure m on XM is called c if it satisfies, for all f for which it makes sense, harmonic (with respect to W) Z c ∆W f dm = 0. f×M c. There exists Let m b be the Γ invariant extension of a harmonic measure m on M c b on M ([Ga]) and, for m-almost b a finite measure m every ξ, a positive harmonic function kξ (x) with kξ (x0 ) = 1 such that the measure m can be written as b m b = kξ (x)(dx × m(dξ)).

The set of harmonic probability measures is a weak* compact nonempty set of measures on XM ([Ga]). A harmonic probability measure m is called ergodic if it is extremal among harmonic probability measures. As a corollary of the results in [L4] and [Z2, Proposition 4.2], we have the following proposition. Proposition 4.1. Let (M, g) be a closed connected Riemannian manifold without focal points and ℓ2 = h. Then, there exists an ergodic harmonic probability measure m on XM c (see below for definitions). b almost all ξ ∈ M such that ∆bξ = ℓ for m

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Indeed, let (M, g) be a closed connected Riemannian manifold. One can consider the f as follows. Fix a point x0 ∈ M f as a reference point. For Busemann compactification of M f f each point x ∈ M , define a function bx (z) on M by f. bx (z) := d(x, z) − d(x, x0 ), ∀z ∈ M

The assignment x 7→ bx is continuous, one-to-one and takes values in a relatively compact f. The set of functions for the topology of uniform convergence on compact subsets of M ˘ of M f is the closure of M f for that topology. The space Busemann compactification M ˘ ˘ (∞) := M ˘ \M f is compact M is a compact separable space. The Busemann boundary M f such that ([LW, Proposition 1]) and is made of 1-Lipschitz continuous functions ξ˘ on M 2 ˘ 0 ) = 0. It is shown in [L4] that if ℓ = h, then there exists an ergodic harmonic ξ(x f× M ˘ by the diagonal probability measure m ˘ corresponding to the quotient of the space M ˘ . (Here, by ∆, we mean the ˘ almost all ξ˘ ∈ M action of Γ such that ∆ξ˘ = ℓ for m Laplacian in the distribution sense.) Note that in case M has no focal points, there exists ˘ 7→ M f ([Z2, Proposition 4.2]) which satisfies ξ˘ = b ˘ for ξ˘ ∈ M ˘. a homemorphism π ˘ :M π ˘ξ Proposition 4.1 follows immediately by letting m be the projection of m ˘ on XM . Proof of Proposition 2.3. Let (M, g) be a closed connected rank one Riemannian manifold without focal points. Assume ℓ2 = h. One can obtain a harmonic measure m satisfying Proposition 4.1 by describing its Γ invariant extension m b ([L4, p. 720]). Set Z dx (4.1) m b t := p(t, x, y)dy , VolM f M

f is the heat kernel on M f. Then pick up m where p(t, x, y), t ∈ R+ , x, y ∈ M b as any ergodic R T 1 decomposition of a limit point of T 0 m b t dt. Its corresponding harmonic measure m on XM satisfies Proposition 4.1 as required.

f(∞) by Theorem 3.12. Hence we Note that the random Brownian path converges at M f×M f(∞). Thus, m is conclude from (4.1) that m b is actually pushed to be supported on M one (and hence is the only one by Theorem 3.1) harmonic measure for the stable foliation. We also conclude from Theorem 3.1 that m is fully supported on SM . f(∞). Moreover, by Proposition 4.1, we have for b is fully supported on M Now we have m b m-a.e. ξ, ∆bξ = ℓ. Recall from section 2.3 that the map x 7→ bξ (x) is of class C 2 and ∆x bξ f(∞). We conclude that the Laplacian of the Busemann depends continuously on ξ ∈ M f, ge) is asymptotically harmonic. function B on SM is constant ℓ. This shows (M 

Remark 4.2. One can also show that any m in Proposition 4.1 is fully supported on SM without using the explicit description of the unique harmonic measure for the stable foliation in Theorem 3.1. Yet, we prefer to use it since it is independent of the assumption of ℓ2 = h, has its own interest in Dirichlet problem and might be useful in a further study of Martin boundary in no focal points case. Here is one approach suggested by Zimmer.

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Any such m also satisfies ([L4]) f×M c. ∇x ln kξ (x) = −ℓ∇x bξ , for m-a.e. b (x, ξ) ∈ M

f(∞) because of the non-differentiability of bξ at ξ ∈ M f b So m-a.e. ξ must belong to M f×M f(∞). By using the density (cf. [Z2, Theorem 5.2]). Thus, m b is supported on M f(∞) (Lemma 3.4), Lemma 3.5 and a similar argument as in of hyperbolic points in M f, of the disintegration of m [Kn1, Lemma 4.1], one can obtain that each m b x, x ∈ M b is fully f supported on M (∞). It follows that m is fully supported on SM . Acknowledgments The first author was partially supported by NSF: DMS 0811127. The second author was partially supported by NSFC: 10901007 and China Scholarship Council; she would also like to thank the department of mathematics of the university of Notre Dame for hospitality during her stay.

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