On the Patterson-Sullivan measure for geodesic flows on rank $1$ manifolds without focal points

In this article, we consider the geodesic flow on a compact rank $1$ Riemannian manifold $M$ without focal points, whose universal cover is denoted by $X$. On the ideal boundary $X(\infty)$ of $X$, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based on the the Patterson-Sullivan measure, we show that the geodesic flow on $M$ has a unique invariant measure of maximal entropy. We also obtain the asymptotic growth rate of the volume of geodesic spheres in $X$ and the growth rate of the number of closed geodesics on $M$. These results generalize the work of Margulis and Knieper in the case of negative and nonpositive curvature respectively.


Introduction and main results
In this article, we study the geodesic flow on a connected closed (compact and having no boundary) rank 1 manifold without focal points. We consider the invariant measure of maximal entropy of the geodesic flow. The uniqueness of such a measure for the geodesic flow on compact manifolds with nonpositive curvature is an important topic in the theory of geodesic flows, which was previously studied by R. Bowen (cf. [8]) and G. Margulis (cf. [29]) in the case of negative curvature, and then proved by G. Knieper for the geodesic flow on a rank 1 manifold with nonpositive curvature (cf. [23]). By following the ideas in [23], we extend Knieper's result to rank 1 manifolds without focal points and prove the existence and uniqueness of the invariant measure of maximal entropy. We will also discuss some related topics, including the distribution of periodic geodesics on a rank 1 manifold without focal points and the asymptotic growth of the volume of geodesic spheres in the universal cover of the manifold.
Suppose that (M, g) is a C ∞ connected compact n-dimensional Riemannian manifold, where g is a Riemannian metric. For each p ∈ M and v ∈ T p M , let γ v be the unique geodesic satisfying the initial conditions γ v (0) = p and γ ′ v (0) = v. The geodesic flow φ = (φ t ) t∈R (generated by the Riemannian metric g) on the unit tangent bundle SM is defined as: Without special indication, all geodesics we are considering in this paper are the geodesics with unit speed.
A φ-invariant probability measure µ is called the measure of maximal entropy if h µ (φ) ≥ h ν (φ) for any φ-invariant probability measure ν. By the variational principle, h µ (φ) = h top (g), where h top (g) denotes the topological entropy of the geodesic flow on SM . In 1970, Margulis constructed a measure of maximal entropy for the geodesic flow on a compact Riemannian manifold of variable negative curvature, in his dissertation at Moscow State University. His results were first published in [29] as a short announcement. Later in [30] he gave more details, and the whole proofs were published eventually in [31]. Using different methods, Bowen also constructed a maximal entropy measure in [6] in 1972, for hyperbolic flows, of which the geodesic flow on a compact manifold of negative curvature is a primary example. Later in [8] he proved that in this case (negative curvature), the maximal entropy measure of the geodesic flow is unique. Therefore, the maximal entropy measures for the geodesic flows constructed by Bowen and Margulis are eventually the same one. Thus we call this measure the Bowen-Margulis measure.
In 1984, A. Katok conjectured that the geodesic flow on a compact rank 1 manifold of nonpositive curvature admits a unique invariant maximal entropy measure (cf. [10]). Here, a rank 1 geodesic is a geodesic which does not have a parallel perpendicular Jacobi field, and a rank 1 manifold is a Riemannian manifold which admits a rank 1 geodesic. More precisely, we present the definition of rank in the following: Definition 1.1. For each v ∈ SM , we define rank(v) to be the dimension of the vector space of parallel Jacobi fields along the geodesic γ v , and rank(M ):=min{rank(v) | v ∈ SM }. For a geodesic γ we define rank(γ)=rank(γ ′ (t)), ∀ t ∈ R.
If M is a rank 1 manifold, the unit tangent bundle SM splits into two invariant subsets: the regular set reg := {v ∈ SM | rank(v) = 1}, and its complement sing := SM \ reg, which is called the singular set. It is still not known if the singular set has zero Liouville volume. This is a wide open problem since 1980's, and the positive answer will imply the ergodicity of the geodesic flow on rank 1 manifolds of nonpositive curvature or without focal points w.r.t. Liouville measure (cf. [40,42]). We also remark that if M has rank greater than 1 and of nonpositive curvature, the celebrated higher rank rigidity theorem, which was established independently by W. Ballmann and K. Burns-R. Spatzier in [2] and [11] based on the work in W. Ballmann-M. Brin-P. Eberlein (cf. [4]) and W. Ballmann-M. Brin-R. Spatzier (cf. [5]), asserts that the universal cover X of M is a flat Euclidean space, a symmetric space of noncompact type, a space of rank 1, or a product of the above types. The higher rank rigidity theorem is also extended to the manifolds without focal points (cf. [39]) and a class of Finsler manifolds (cf. [41]).
Katok's conjecture was eventually proved by Knieper in [23]. In his proof, instead of considering the measures supported on closed geodesics as in [6], Knieper studied the socalled Busemann density defined by using the Poincaré series, and used it to construct the measure of maximal entropy (we call it Knieper measure). Knieper's innovative work points out a new way to study the measure of maximal entropy for geodesic flows. An immediate question is that does this result hold for more general situations? The manifolds without focal points / conjugate points are usually considered to be the natural extension of the conception of manifolds with nonpositive curvature. Definition 1.2. Let γ be a geodesic on (M, g). A pair of distinct points p = γ(t 1 ) and q = γ(t 2 ) are called focal if there is a Jacobi field J along γ such that J(t 1 ) = 0, J ′ (t 1 ) = 0 and d dt J(t) 2 | t=t 2 = 0; p = γ(t 1 ) and q = γ(t 2 ) are called conjugate if there is a nonidentically-zero Jacobi field J along γ such that J(t 1 ) = 0 = J(t 2 ).
A compact Riemannian manifold (M, g) is called a manifold without focal points / without conjugate points if there is no focal points / conjugate points on any geodesic in (M, g).
From the definitions, it is easy to see that if a manifold has no focal points then it has no conjugate points. It is well known that all manifolds with nonpositive curvature have no focal points. In this sense, manifolds without focal / conjugate points can be regarded as a generalization of the manifolds with nonpositive curvature. This generalization is non-trivial, since it is easy to construct a compact manifold without focal points whose curvature is not everywhere nonpositive (cf. [18]).
In this paper we generalize Knieper's work and prove the existence and uniqueness of the measure of maximal entropy on a compact rank 1 manifolds without focal points, by employing Knieper's idea to the more general situation. This is the following theorem: Remark 1. In a recent preprint [17], K. Gelfert and R. Ruggiero proved that the geodesic flow on a compact surface without focal points has a unique measure of maximal entropy. Their idea is to consider the time-preserving semi-conjugacy from the geodesic flow to a continuous expansive flow with a local product structure, which has a unique measure of maximal entropy.
The property h top (φ| sing ) < h top (g) is usually called the "entropy gap" for the geodesic flows. It is obtained for the geodesic flows on compact rank 1 manifolds of nonpositve curvature by Knieper in [23]. Recently, this is generalized to a "pressure gap" for the geodesic flows on compact rank 1 manifolds of nonpositve curvature, by using a different method by K. Burns-V. Climenhaga-T. Fisher-D. J. Thompson in [9]. Our result shows the entropy gap in the situation of no focal points.
The key ingredient in the proof is the Patterson-Sullivan measure constructed from the Poincaré series. Let M = X/Γ be a compact rank 1 manifold without focal points, where X is the universal cover of M and Γ is a discrete subgroup of the isometry group Iso(X). We will extend the Patterson-Sullivan construction to the rank 1 manifolds without focal points, and show that the Patterson-Sullivan measure is essentially the unique Busemann density. We will discuss the notion of Busemann density and the Patterson-Sullivan construction in Subsection 3.1.
We note that in nonpositive curvature case, Knieper's method relies on the convexity properties of various distance functions. In no focal points case, convexity property are replaced by sort of the no maxima property. Nevertheless, some results such as flat strip lemma remain true. In the course of the proof of Theorem A, we establish all the geometric properties needed in Knieper's method, more precisely Propositions 2.3, 2.6, 2.8-2.11 and 2.14. In Propositions 2.3 and 2.6, we prove the continuity property of geodesics at infinity and describe the property of cones of simply connected manifolds without focal points; in Proposition 2.8 we show that compact manifolds without focal points satisfy the duality condition, a property that has many important applications, including the rank rigidity theory in nonpositive curvature(cf. [3,14]); in Propositions 2.9-2.11, we establish the existence of the connecting geodesics between the neighborhoods of the endpoints of a rank 1 geodesic, which is crucial for the semi-local properties in the no focal case appearing in Subsection 3.2; in Proposition 2.14, we prove that the fundamental group acts minimally on the ideal boundary, which is extraordinarily important in our discussion on the Patterson-Sullivan measure (see Proposition 3.22). These properties are important for the study of the dynamics of the geodesic flows on rank 1 manifolds with no focal points, and hence of independent interests. Recently in [27] we used these properties to prove that the geodesic flows on compact rank 1 manifolds without focal points are topologically transitive.
Theorem B. Let M = X/Γ be a compact rank 1 manifold without focal points, then up to a multiplicative constant, the Busemann density is unique, i.e., the Patterson-Sullivan measure is the unique Busemann density.
The Busemann density was first constructed by Patterson in [34]. Uniqueness of the Busemann density was established by Sullivan (cf. [38]) for compact manifolds of negative curvature and by Knieper (cf. [22]) for compact rank 1 manifolds of nonpositive curvature. To our knowledge, it is the first time the Busemann density is considered for compact rank 1 manifold without focal points, in Theorem B. We also show that the critical exponent of the Poincaré series of the co-compact group Γ ⊂ Iso(X) with M = X \ Γ coincide with the topological entropy of the geodesic flow on SM .
Since the Poincaré series is of divergent type, the Patterson-Sullivan measure can be used to construct a finite measure on SX which is invariant under the geodesic flow and Γ-action. The main work in this article is to show the invariant measure under the geodesic flow on SM constructed in this way is the unique measure of maximal entropy, proving Theorem A.
Furthermore, the power of the Patterson-Sullivan measure is not only limited to this problem. It can also be used to investigate the asymptotic geometry of the rank 1 compact manifolds without focal points. Let B(x, r) be a geodesic ball in X about x with radius r > 0, and S(x, r) = ∂B(x, r) be a geodesic sphere in X. Let h = h top (g) be the topological entropy of the geodesic flow, which will be shown to coincide with the critical exponent of the Poincaré series of the co-compact group Γ ⊂ Iso(X) with M = X \ Γ, as mentioned above. The classical results of Manning (cf. [28]) and Freire-Mañé (cf. [16]) show h = lim r→∞ log Vol(B(x, r)) r .
In [29], Margulis obtained a finer estimate in the case of negative curvature: there are constants a > 0 and r 0 > 0 such that It is Knieper who applied the Patterson-Sullivan measure to obtain a same result on the asymptotic growth of the volume of the geodesic spheres for compact rank 1 manifolds of nonpositive curvature (cf. [22]). His method can be extended to compact rank 1 manifolds without focal points: Theorem C. Suppose (M, g) is a compact rank 1 manifold without focal points and X is its universal cover. Let x ∈ X be an arbitrary point and S(x, r) be the sphere centered at x with radius r. Then there are constants a > 0 and r 0 > 0 such that A straightforward application of Theorem A is estimating the distribution of the regular and singular primitive closed geodesics with a given upper-bound of the periods on a compact rank 1 manifold without focal points. Here, we say a closed geodesic is primitive, if it is not an iterate of another closed geodesic. If a closed geodesic staying in the regular set, we call it a regular closed geodesic, otherwise we call it a singular closed geodesics. Let P(M ) be a maximal set of geometrically distinct primitive closed geodesics which represent different free homotopy classes. For each constant t > 0, we use P(t) to denote the set of geodesics in P(M ) with least periods less than or equal to t. Note that since the flat strip theorem holds for manifolds without focal points, the least periods of all closed geodesics are equal in a given free homotopy class. Let P reg (t) ⊂ P(t) be the subset of regular closed geodesics in P(t), and P sing (t) = P(t)\P reg (t) be the subset of singular closed geodesics in P(t). Denote by P reg (t) := ♯P reg (t) and P sing (t) := ♯P sing (t). Then Theorem A (2) and Theorem C imply the following result: Theorem D. Let (M, g) be a compact rank 1 Riemannian manifold without focal points. Then there exist a > 0 and t 1 > 0 such that for all t > t 1 , Moreover, there exist positive constants ǫ and t 2 such that This paper is organized in the following way: In Section 2, we study the geometric properties of rank 1 manifolds without focal points, which will be frequently used in our subsequent discussion. We present the Patterson-Sullivan construction for rank 1 manifolds without focal points and prove Theorem B in Section 3. A key technical result about the Patterson-Sullivan measure is prepared in Subsection 3.2. Then Theorem C is proved in Sections 4. In the last two Sections 5 and 6, we prove Theorems A and D. We should point out that our main arguments follow the ideas of Knieper's work in [22] and [23], which are showed to work for geodesic flows on rank 1 manifolds without focal points, based on the geometric properties we establish in this paper.
2 Geometric properties of rank 1 manifolds without focal points In this section, we present some geometric results on rank 1 manifolds without focal points, which will be used in the following discussions. We remark that, although in this paper we only discuss the rank 1 manifolds without focal points, some of the results are also valid in more general situations.
Let X be the universal covering manifold of M and d is the distance function on X induced by the lifted Riemannian metricg on X. Suppose h 1 and h 2 are both geodesics in X. We call h 1 and h 2 are positively asymptotic if there is a positive number C > 0 such that We say h 1 and h 2 are negatively asymptotic if (2.1) holds for all t ≤ 0. h 1 and h 2 are said to be biasymptotic if they are both positively asymptotic and negatively asymptotic. The relation of (positive / negative) asymptoticity is an equivalence relation between geodesics on X. The class of geodesics that are positively / negatively asymptotic to a given geodesic γ is denoted by γ(+∞) / γ(−∞) respectively. We call them points at infinity. Obviously, γ v (−∞) = γ −v (+∞). We use X(∞) to denote the set of all points at infinity, and call it the boundary at infinity, or the ideal boundary.
Let X = X ∪ X(∞). For each point p ∈ X and v ∈ S p X, each points x, y ∈ X − {p}, positive numbers ǫ and r, and subset A ⊂ X − {p}, we define the following notations: • γ p,x is the geodesic from p to x and satisfies γ p,x (0) = p.
T C(v, ǫ, r) is called the truncated cone with axis v and angle ǫ. Obviously γ v (+∞) ∈ T C(v, ǫ, r). There is a unique topology τ on X such that for each ξ ∈ X(∞) the set of truncated cones containing ξ forms a local basis for τ at ξ. This topology is usually called the cone topology. Under this topology, X is homeomorphic to the closed unit ball in R dim(X) , and the ideal boundary X(∞) is homeomorphic to the unit sphere S dim(X)−1 . For more details about the cone topology, see [15] and [14].
The following results of manifolds without focal points are well known. We state them as two lemmas here, for these results will be used very often in this paper. [32]). Let X be a simply connected Riemannian manifold without focal points.
1. Let h 1 and h 2 be a pair of distinct geodesic rays starting from a same point in X.
2. Let h 1 and h 2 be a pair of positively asymptotic geodesics. Then the distance function Remark 2. Lemma 2.1 implies that on a simply connected Riemannian manifold without focal points, two distinct geodesics can not be positively or negatively asymptotic if they cross at some point on X.
Lemma 2.2 (cf. for example Proposition 2.8 in Katok [19]). Let h 1 and h 2 be two distinct geodesics on a simply connected Riemannian manifold without focal points, then for any a ∈ R and t ∈ [0, a], the following inequality holds: Our first result in this section is the property of the continuity to infinity. We define a metric δ on SX by δ(v, w) := d(πv, πw) + d(exp v, exp w).
It can be shown that this distance is equivalent to the distance induced by the standard Sasaki metric. The following proposition shows the continuity of the map (v, t) → γ v (t) from SX × [−∞, +∞] to X with respect to the metric δ, on a simply connected manifold without focal points. It is the no focal points version of Proposition 2.13 of [15].
3 (Continuity to infinity). Let X be a simply connected manifold without focal points, the map is continuous.
Let q n = γ vn (t n ) = γ wn (s n ). Now we use γ v −1 n to denote the geodesic from q n to p with γ v −1 n (0) = q n , and γ qn to denote the geodesic from q n to p n with γ qn (0) = q n , respectively. By Lemma 2.1 and (2.2), we have Thus By v n → v and the above inequality, we get w n → v. This implies that for any ǫ > 0 there is an N = N (ǫ) > 0 such that if n > N , then γ vn (t n ) ∈ T C(v, ǫ, sn 2 ). So it follows that, under the cone topology, γ vn (t n ) = γ wn (s n ) → γ v (+∞). Now we consider the continuity of Ψ(·, +∞) : SX → X(∞). Suppose {v n } ⊂ SX is a convergent sequence with lim n→∞ v n = v ∈ S p X for some p ∈ X. For each γ vn (+∞) ∈ X(∞), there is a unique vector w n ∈ S p X such that γ wn (+∞) = γ vn (+∞). Similar as the argument in the above we can show that w n → v. We know on the compact space S p X, all the metrics are equivalent, thus the metric δ defined above is equivalent to the usual angle metric. Therefore lim n→+∞ θ n = 0, where θ n is the angle between v and w n . Thus, passing to a subsequence if necessary, we can assume that θ n < 1 n . This means that γ vn (+∞) = γ wn (+∞) ∈ T C(γ v (+∞), 1 n , n). Since The following proposition is a corollary of Proposition 2.3. It will be used in our discussion in the next section.
Proposition 2.4. Let X be a simply connected manifold without focal points. Then for each point p ∈ X, the map is a homeomorphism.
Proof. By Lemma 2.1(1) we know that the map γ +∞ is bijective. Proposition 2.3 implies that γ +∞ is a continuous map from S p X to X(∞). Thus it suffices to show that the inverse map γ −1 +∞ : X(∞) → S p X is continuous.
For each p ∈ X, let B p = {v ∈ T p X | v ≤ 1} ⊂ T p X, which is the unit ball in T p X. Using the same method in the proof of Proposition 2.4, we can get the following result. Corollary 2.5. Let X be a simply connected manifold without focal points. Then for each point p ∈ X, the map is a homeomorphism.
Proposition 2.6. Let X be a simply connected manifold without focal points. Then for any v ∈ SX, R > 0, and ǫ > 0, there is a constant L = L(v, ǫ, R) such that for all t > L, Proof. We prove this proposition by contradiction. Suppose that there is a vector v ∈ SX, and constants ǫ > 0 and R > 0 such that ∀L > 0, we can always find t > L satisfying Let p be the footpoint of v, i.e. v ∈ S p X. Take L n = n, then ∃ t n > L n = n, and i.e. we have t n − R < d(p, p n ) < t n + R.
Let γ n be the geodesic ray from p to p n and v n = γ ′ n (0). Then by Lemma 2.1, it's easy to see that However, by the inequality (2.3) and Proposition 2.3, we have γ v∞ (+∞) = γ v (+∞). This contradicts to Lemma 2.1 since both γ v∞ and γ v start from p.
From the argument in the above, we know that for any v ∈ SX, R > 0, and ǫ > 0, there is a constant L such that for all t > L, B(γ v (t), R) ⊂ C(v, ǫ). We are done with the proof.
In the following, we prove the duality property on Riemannian manifolds without focal points. Before stating the result, we should introduce the concepts of recurrent point and nonwandering point with respect to a discrete subgroup Γ of isometry group Iso(X) of X, where X is a simply connected manifold.
We use Ω(Γ) to denote the set of all nonwandering points of geodesic flow φ t : SX → SX.
This definition of Ω(Γ) was introduced by W. Ballmann in [1]. Let C : X → M be the universal covering map. Then dC(Ω(Γ)) is the nonwandering set of the geodesic flow φ t : SM → SM in the common sense in dynamical systems ( [14,20,33]). Moreover, it's easy to verify that for any α ∈ Γ, the following diagrams commute. Here we use φ t to denote both the geodesic flow on SX and the geodesic flow on SM .
Proposition 2.8 (Duality property). Let X be a simply connected manifold without focal points and Γ be a discrete subgroup of the isometry group Iso(X).
Proof. First we show that Ω(Γ) = SX. We prove this by contradiction. Assume that there exists a vector v ∈ SX such that v / ∈ Ω(Γ). Then we can find a neighborhood U of v in Together with the second diagram in (2.4), we know that Since C is an Inj(M )-isometry (see for example [33] remark 3.37) and the geodesic flow preserves the Liouville measure λ on SM , we know that for any t ∈ R, (2.7) and (2.8) imply that φ nA (V ) (n = 1, 2, 3, ...) are pairwisely disjoint sets with equal positive Liouville measure in SM . While SM is compact, λ(SM ) = 1 < +∞, this is impossible. From the argument in the above, we can conclude that Ω(Γ) = SX.
This implies that α n γ vn (t n ) → γ v (0), and then Therefore by the property of continuity to infinity (Proposition 2.3), we have We have done the proof of the first limit.
For the second one, let γ n (t) = α n γ vn (t n − t), then Take p = γ v (0), then (2.5) is valid. Proposition 2.6 implies that if (2.5) holds for one point, then it holds for all points in X. This complete the proof of Proposition 2.8.
The following proposition was first discovered by J. Watkins in [39]. Here we give a new proof by using a different method. This proposition can be used as a criterion to distinguish a vector of higher rank on a simply connected manifold without focal points. Proposition 2.9. Suppose X is a simply connected manifold without focal points and v ∈ SX. If there is a constant c > 0 such that for all k ∈ Z + , we can always find a pair of points in the cones: Proof. First of all, we will show that under the assumption of this proposition, both of the following equalities hold: lim As a first step to prove the above result, we show that: • At least one of the above equalities holds. i.e. we can not find a subsequence {i k } ∞ k=1 ⊂ Z + which makes both (2.9) and (2.10) false.
Suppose the claim is not true, i.e., there exists a positive number A, and subsequences By passing a subsequence if necessary, this implies that p i k converges to some point Moreover, by the continuity of the distance function d(x, y), there is a This implies that when k > max{K 1 , K 2 }, we have We use γ i k to denote the geodesic from p i k to q i k satisfying γ i k (−t 1 ) = p i k , then by Lemma 2.2, we have Specifically, let t = 0, we get a contradiction to the assumption d(γ v (0), γ p k ,q k ) ≥ c. This proves the claim, i.e. at least one of the equalities (2.9) and (2.10) holds.
The second step is to prove that for some A > 0.
This will lead to the equalities (2.9).
which also contradicts to the assumption d(γ v (0), γ p k ,q k ) ≥ c. We have done with the proof of equality (2.9).
The equality (2.10) can be proved in a similar way, so we omit the proof. In summary, both of (2.9) and (2.10) hold, thus In order to prove Proposition 2.9, we consider the following two cases: • Case I there exists a positive constant R such that where γ k is the geodesic connecting p k and q k .
We choose the parametrization of γ k such that d( for all t ∈ R, and by Proposition 2.3, we have This leads to the existence of a flat strip bounded by γ v and γ v ′ (cf. [32]). Thus rank(v) ≥ 2.
is an increasing function of s. Let γ k,s be the geodesic connecting c k (s) and b k (s) for each s ∈ [0, 1], with the parametrization γ k,0 (0) = γ v (0) and γ k,s (0) is a smooth curve of s.
Using the same argument in the beginning of the proof of this proposition, we have Then by the fact By Proposition 2.3, we know that Since X has no focal points, it's impossible that both of b k (s k ) and c k (s k ) are on the . This implies that γ v and γ v∞ bound a flat strip, so rank(v) ≥ 2. We complete the proof of this proposition.
The following two propositions are corollaries of Proposition 2.9.
Proposition 2.11. Let X be a simply connected manifold without focal points and rank(v) = 1, where v ∈ SX is a unit vector. Then for any ǫ > 0, there are neighborhoods U ǫ of Proof. By Proposition 2.10, for sufficiently small neighborhoods U ǫ , V ǫ and all ξ ∈ U ǫ , η ∈ V ǫ , there is also a geodesic γ ξ,η connecting ξ and η.
Then by a similar argument of Proposition 2.9, we can show that rank(v) ≥ 2, which contradict our assumption that rank(v) = 1.
We choose the parametrization of γ n such that d( Since {γ n (0)} are in a compact subset, passing to a subsequence if necessary, we consider the following two cases: by Lemma 2.1 and Proposition 2.3, we have lim n→+∞ γ ′ n (0) = γ ′ v (0) = v. Since higher rank set sing is closed, rank(v) ≥ 2, a contradiction.
An isometry α ∈ Iso(X) is called axial if there exists a geodesic γ and a t 0 > 0 such that for any t ∈ R, α(γ(t)) = γ(t + t 0 ). Correspondingly γ is called an axis of α. Axial isometry is abundant for compact manifolds without conjugate points. Each element in Γ is axial if X/Γ is compact (cf. [12] Lemma 2.1). The following property of axial isometry was proved by Watkins: Lemma 2.12 (cf. Watkins [39]). Let X be a simply connected manifold without focal points and let γ be a rank 1 geodesic axis of Lemma 2.12 and Proposition 2.11 imply the following results.
Corollary 2.13. Let X be a simply connected manifold without focal points, then an end point of a rank 1 axis can be connected by rank 1 geodesics to all the other points in X(∞). This implies that, for every ξ ∈ X(∞), there is a rank 1 geodesic γ + with γ + (+∞) = ξ and a rank 1 geodesic γ − with γ − (−∞) = ξ.
The connecting geodesic in Corollary 2.13 is unique since it is rank 1.
Combining Proposition 2.8 and Lemma 2.12, we can get the following result. We remark that this result is the no focal points version of Lemma 2.3 and Corollary 2.4 of [24].
Proposition 2.14. Let X be a simply connected manifold without focal points and let Γ ⊂ Iso(X) be such that M = X/Γ is compact. If γ is a rank 1 geodesic on X, then there exists a sequence of rank 1 axes γ n of deck transformations α n ∈ Γ such that γ n → γ. Furthermore, Γ acts minimally on X(∞), i.e., for any ξ ∈ X(∞), Γξ = X(∞).
Proof. By Proposition 2.8, there is a sequence {α n } +∞ n=1 ⊂ Γ such that (2.11) Lemma 2.12 implies that the closure of sufficiently small neighborhoods U n of γ(−∞) and are invariant under α n and α −1 n respectively. Since in cone topology, X = X ∪ X(∞) is homeomorphic to the dim(X)-unit ball B dim(X) in R dim(X) , we know U n and V n are both homeomorphic to B dim(X) . Then by the Brouwer fixed point theorem, α n has one fixed point in p n ∈ U n and α −1 n has one fixed point in q n ∈ V n .
We claim that both p n and q n belong to X(∞) for sufficiently large n. We prove this fact by contradiction. First, if p n ∈ X and q n ∈ X, then α n = id since α n ∈ Γ ⊂ Iso(X), which contradicts to (2.11). Second, if p n ∈ X and ξ = q n ∈ X(∞), then α n maps the geodesic ray γ pn,ξ to itself and fixes endpoints, thus α n = id, which also contradicts to (2.11). The case p n ∈ X(∞) and q n ∈ X can be considered in the same way. Thus both p n and q n belong to X(∞) for large enough n.
Without loss of generality, we assume that both p n and q n belong to X(∞) for all n ≥ 1. By the definition of U n and V n , Proposition 2.11 implies that for large enouth n, the connecting geodesic γ n = γ pn,qn is the rank 1 axis of the axial isometry α n . Furthermore, γ n (−∞) = p n → γ(−∞), γ n (+∞) = q n → γ(+∞), thus γ n → γ. Now we will prove Γξ = X(∞). We only need to show that for any η ∈ X(∞) with η = ξ, there exists a sequence {β n } ⊂ Γ such that β n (ξ) → η. It's easy to see that the rank function is upper semi-continuous and integer-valued, thus the set of vectors v ∈ SM such that rank(w) = rank(v) for all w sufficiently close to v is dense in SM . Since this set is exactly the set of rank 1 vectors (cf. [39]), it follows that the set of rank 1 vectors is dense in SM . Therefore the set of rank 1 vectors is also dense in SX. The above result implies that the set of vectors tangent to rank 1 axes are dense in SX. Therefore, for any open neighborhood U ⊂ X(∞) of η, there is a rank 1 axis γ v of some α ∈ Γ, satisfying that γ v (+∞) ∈ U and γ v (−∞) = ξ. Then by Lemma 2.12 there exists n ∈ Z + such that α n ξ ∈ U . Since U can be chosen arbitrarily small, we can find a sequence {n k } ⊂ Z + and {α k } ⊂ Γ such that lim k→+∞ α n k k ξ = η.
3 Existence and uniqueness of the Busemann density

Construction of a Busemann density
For each pair of points (p, q) ∈ X × X and each point at infinity ξ ∈ X(∞), the Busemann function determined by p, q and ξ is where γ p,ξ is the geodesic ray from p to ξ. The Busemann function b p (q, ξ) is well-defined since the function t → d(q, γ p,ξ (t)) − t is bounded from above by d(p, q), and decreasing in t (this can be checked by using the triangle inequality).
The level sets of the Busemann function b p (q, ξ) are called the horospheres centered at ξ. We denote by H ξ (γ p,ξ (t)) the horosphere centered at ξ and passing through γ p,ξ (t). For more details of the Busemann functions and horosphpers, please see [13,35,36]. At here, we restate one important property of the horospheres, which will be used later.
Definition 3.16. Let X be a simply connected Riemannian manifold without conjugate points and Γ ⊂ Iso(X) be a discrete subgroup. For a given constant r > 0, a family of finite Borel measures {µ p } p∈X on X(∞) is called an r-dimensional Busemann density if 1. For any p, q ∈ X and µ p -a.e. ξ ∈ X(∞), where b p (q, ξ) is the Busemann function.
The construction of such a Busemann density is due to Patterson [34] in the case of Fuchsian groups, and generalized by Sullivan [38] for hyperbolic spaces.
Let X be a simply connected Riemannian manifold without conjugate points and Γ ⊂ Iso(X) be a infinite discrete subgroup. For each pair of points (p, p 0 ) ∈ X × X and s ∈ R, Poincaré series is defined as This series may diverge for some s > 0. The number δ := inf{s ∈ R | P (s, p, p 0 ) < +∞} is called the critical exponent of Γ. By the triangle inequality, it is easy to check that δ is independent of p and p 0 . We say Γ is of divergent type if the Poincaré series diverges when s = δ. It is well-known that whether Γ is of divergent type or not is independent of the choices of p and p 0 . The following lemma shows the relation between the critical exponent and the topological entropy of the geodesic flow on M = X/Γ. Proof. Since X is the universal cover of M , by Freire and Mañé's Theorem (cf. [16]), we know that where h(g) := lim r→+∞ log VolB(p, r) r is the volume entropy of the geodesic flow. Here B(p, r) denotes the ball in X centered at p ∈ X with radius r > 0. In [28], A. Manning showed that this limit is independent of p.
Let d and A be the diameter and area of a fundamental domain F respectively. Then there exists K 1 ∈ Z + such that for any k ≥ max{K 0 , K 1 }, .
Then it follows that δ = lim On the other hand, we claim that there exists k i → ∞ such that Assume not. Then there exists K 2 ∈ Z + , such that for any k > K 2 ,

Now observe that
If we let s = h(g)−2ǫ and use the claim above, then the right side of the above inequality is infinity. Thus δ ≥ h(g) − 2ǫ. (3.14) So δ = h(g) = h top (g) by (3.13) and (3.14).
Here in constructing a Busemann density, we consider the case that the group Γ is of divergent type first. At the end of this subsection, we give a sketch of Patterson's method to deal with the convergent type (cf. [34]). Now for p ∈ X, consider a weak ⋆ limit lim s k ↓h µ p,s k = µ p .
Since P (s, p 0 , p 0 ) is divergent for s = h and Γ is discrete, it's obvious that supp(µ p ) ⊂ Γp 0 ∩ X(∞). In fact, one can easily check that lim s k ↓h µ p,s k (A) = 0 for all bounded open sets A ⊂ X, therefore supp(µ p ) ⊂ X(∞). Moreover, we can show that {µ p } p∈X is an h-dimensional Busemann density. We should emphasis that the different choices of {s k } may lead different weak ⋆ limits. But in the case of rank 1 manifolds without focal points, we will show that the the Busemann density is unique, i.e., independent of {s k }.
In order to show {µ p } p∈X constructed above is an Busemann density, we need the following two results about Busemann function. Proposition 3.18. For each pair of points p, q ∈ X, the map is continuous.
Proof. We know that if t = b p (q, ξ) for some t ∈ R, then the horosphere passing through q and centered at ξ will intersect the geodesic γ p,ξ perpendicularly at γ p,ξ (−t) (cf. [36] Lemma 4.2). Proposition 2.4 implies that for any p ∈ X, and any {ξ n } ⊂ X(∞) with lim n→+∞ ξ n = ξ, we have lim n→+∞ γ ′ p,ξn (0) = γ ′ ξ (0). Therefore, if we fix a point p ∈ X, then the geodesic γ p,ξ depends continuously on ξ, Moreover, Theorem 3.15 tells us that, for the manifold without focal points, the horospheres depend continuously on their centers. This implies that the intersection points of the geodesic γ p,ξ and the horosphere H ξ (q) depends continuously on ξ ∈ X(∞). Then by our explanation in the last paragraph, we know the Busemann function b p (q, ξ) depends continuously on ξ.
Proposition 3.19. For each pair of points p, q ∈ X, if there is a sequence {x n } ⊂ X with lim n→+∞ x n = ξ, then lim n→+∞ {d(q, x n ) − d(p, x n )} = b p (q, ξ).
Proof. Since lim n→+∞ x n = ξ, then by passing to a subsequence if necessary, we can assume that x n ∈ T C(v, 1 n , n), where v = γ ′ p,ξ (0) and T C(v, 1 n , n) is the truncated cone introduced in Section 2. Let v n = γ ′ p,xn (0), then we know that the angle between v and v n is smaller than 1 n . This implies lim n→+∞ v n = v. Let ξ n = γ vn (+∞), by Proposition 2.4, lim n→+∞ ξ n = ξ.
Fix a point ξ ∈ X(∞), let a = b p (q, ξ). By Proposition 3.18, lim n→+∞ ξ n = ξ implies that for any ǫ > 0, there exists N ∈ Z + , such that for all n > N , we have Since the function t → d(q, γ w (t)) − t is nonincreasing, we know that for any ǫ > 0, there exists a constant T ∈ R, such that Now we fix the number T . Since lim n→+∞ v n = v, we know that there exists N 1 ≥ max{T, N }, such that , ∀ t ∈ [0, T ] and n > N 1 .
Thus ∀n > N 1 , we have By the nonincreasing property of the function t → d(q, γ w (t)) − t, we know that for all n ≥ N 1 , We are done with the proof. Proof. We shall prove that {µ p } p∈X satisfies the two properties in the Definition 3.16.
When Γ is of convergent type, Patterson showed in [34] that one can weight the Poincaré series with a positive monotone increasing function to make the refined Poincaré series diverge at s = h. More precisely, we consider a positive monotonely increasing function g : R + → R + with lim t→+∞ g(x+t) g(t) = 1, ∀x ∈ R + , such that the weighted Poincaré series P (s, p, p 0 ) := α∈Γ g(d(p, αp 0 ))e −s·d(p,αp 0 ) diverges at s = δ. For the construction of the function g, see [34] Lemma 3.1. Then, for the weighted Poincaré series P (s, p, p 0 ), we can always construct a Busemann density by using the same method as above.

Projections of geodesic balls to X(∞)
In this subsection, we will show some semi-local properties of a Busemann density {µ p } p∈X . These properties will play key roles in our proof of main theorems in subsequent sections.
For A ⊂ X and p ∈ X, Proposition 3.23. The projection map is continuous.
The following important semi-local properties were first discovered by Knieper [22] (see also Knieper [23,24,25]) in the setting of rank 1 manifolds with nonpositive curvature. Based on the results in Section 2 and Subsection 3.1, we can extend them to rank 1 manifolds without focal points by following Knieper's argument.
Proposition 3.24. Let X be a simply connected rank 1 manifold without focal points which has a compact quotient, and {µ p } p∈X be an h-dimensional Busemann density on X(∞). Then there exist positive constants R and l such that for all r > R, (1) µ p (pr x B(p, r)) ≥ l, for all p ∈ X and x ∈ X, where B(p, r) is the open ball of radius r, centered at p; (2) there is a positive constant a = a(r) such that for any x ∈ X \ {p} and ξ = γ p,x (−∞), we have 1 a · e −h·d(p,x) ≤ µ p (pr ξ B(x, r)) ≤ a · e −h·d(p,x) ; (3) there is a positive constant b = b(r) such that for any x ∈ X, Proof. We prove item (1) first. Items (2) and (3) are consequences of (1). We already know, by Proposition 3.22, that supp(µ p ) = X(∞) for all p ∈ X. This means that for any h-dimensional Busemann density µ p and any open set U ⊂ X(∞), µ p (U ) > 0. Specifically, for any v ∈ SX and ǫ > 0, we have is defined in Section 2.
Now we consider the projection pr ξ : X → X(∞), for ξ ∈ X(∞). We have the following lemma: Lemma 3.25. Suppose ξ ∈ X(∞). Then for any x ∈ X there is an R = R(x) > 0 such that for any r > R, pr ξ (B(x, r)) contains an open subset of X(∞).
Next we study the projection pr x : X → X(∞), for general x ∈ X = X ∪ X(∞). Fix a compact fundamental domain F ⊂ X and a point x 0 ∈ F. Consider the set pr x (B(p, r)) where p ∈ F. Combining Lemma 3.25 with Proposition 3.23, we get the following conclusion: Lemma 3.26. There exist positive constants ǫ and R, such that for any p ∈ F and any The proof of this lemma on rank 1 manifolds with nonpositive curvatures can be found in Knieper [22], p. 762. Based on Lemma 3.25, on can check that the above result also holds for rank 1 manifolds without focal points, by following the same argument. So we omit the proof here. Now we claim that, for any ǫ > 0, there is an l = l(ǫ) > 0 such that for all p ∈ F and v ∈ S x 0 X, we have µ p (C ǫ (v)) > l. This can be proved by contradiction. We know that since supp(µ p ) = X(∞), then µ p (C ǫ (v)) > 0, for all p ∈ X, v ∈ S x 0 X and ǫ > 0. Assume that there is a number ǫ 0 > 0, and sequences {v n } ⊂ S x 0 X and {p n } ⊂ F with v n → v, such that µ pn (C ǫ 0 (v n )) < 1 n . Since v n → v, there is a constant N > 0 such that if n > N , then C ǫ 0 (v n ) ⊃ C ǫ 0 2 (v). Thus for any p ∈ F, we have This implies that µ p (C ǫ 0 2 (v)) = 0, which is a contradiction.
Based on the discussion in the above, we know that there exist R > 0 and l > 0 such that if r > R, then for all x ∈ X and p ∈ F. In general, for any p ∈ X we can always find an isometry β ∈ Γ such that βp ∈ F. Then by the Γ-invariance of µ p , for all r > R, we have µ p (pr x B(p, r)) = µ βp (pr βx B(βp, r)) > l.
We are done with the proof of (1). Here the last inequality follows from Proposition 3.24(1).

Now we prove statement (2). It is obvious that for all
Moreover, one can check that for all p ∈ X and η ∈ pr ξ B(x, r), This implies that b x (p, η) + 2r ≥ d(x, p). Then we have where e (h+1)K is the uniform upper bound of {µ p } p∈X in Proposition 3.21. Let a(r) = max{ 1 l(r) , e (h+1)K e 2rh }, we get 1 a · e −h·d(p,x) ≤ µ p (pr ξ B(x, r)) ≤ a · e −h·d(p,x) . We are done with the proof of (2). Statement (3) can be proved in the same way, so we omit its proof here.

Uniqueness of the Busemann density
We restate Theorem B below. It asserts the uniqueness of the Busemann density, which was originally proved by Knieper in [22] under the assumption of nonpositive curvature. One can check that, based on our discussion in the above, Knieper's proof also works in the situation of no focal points.
Theorem B. Let M = X/Γ be a compact rank 1 manifold without focal points, then up to a multiplicative constant, the Busemann density is unique, i.e., the Patterson-Sullivan measure is the unique Busemann density.
We sketch the idea of the proof, following closely [22] In the proof of the above estimations, Knieper used the property that the distance function of two geodesics on a simply connected nonpositively curved manifold M (usually we call it a Hadamard manifold) is convex, i.e., if c 1 and c 2 are two unit speed geodesics on a Hadamard manifold M , then the function t → d(c 1 (t), c 2 (t)) is a convex function. Although the convexity is no longer valid for manifolds without focal points, we can still get the same estimations based on Lemma 2.1. Then we can prove that the volume entropy h equals the Hd(X(∞)), i.e., the Hausdorff dimension of X(∞). Finally, we can show that all Busemann densities have to coincide up to a constant.
A natural question is to study the existence and unqueness of Busemann density for rank 1 manifolds without conjugate points. In our proof of Busemann density for rank 1 manifolds without focal points, Proposition 3.24 (1) plays a key role, which in turn follows from the geometric properties stated in Lemma 2.12 and Proposition 2.11. However, we don't know whether Lemma 2.12 and Proposition 2.11 are still true under the assumption of no conjugate points.

Exponential volume growth rate of geodesic spheres
In this section, we consider the growth rate of the geodesic sphere S(x, r) about x with radius r, on the universal cover X of a compact rank 1 manifold M without focal points. Based on the technical results prepared in Proposition 3.24, we prove Theorem C in this section. We remark that it is a generalization of Theorem 5.1 in [22].
Proof of Theorem C. Since we are interested in only sufficiently large r, we can assume r > 3R, where R is the constant in Proposition 3.24. Let {q 1 , q 2 , · · · , q k } be a maximal 3R−separated set of S(x, r). Then we have that and · e −h·r ≤ µ x (pr x B(q i , ρ)) ≤ b · e −h·r , i = 1, 2, · · · , k.
Notice that X(∞) = pr x (S(x, r)). Combining with (4.15), we have that Also by (4.16), we know that From the two inequalities in the above we know that there is a constant c > 0 such that the following inequality holds 1 c · e h·r ≤ k ≤ c · e h·r .
Since M is a compact manifold, we can see that there is a positive constant A, such that for all r > 3R and R ≤ ρ ≤ 3R, we have 1 A ≤ Vol(B(q i , ρ) ∩ S(x, r)) ≤ A, i = 1, 2, · · · , k.
Therefore, we get 1 cA ≤ VolS(x, r) e h·r ≤ cA.
Let a = cA > 0 and r 0 = 3R, then we are done with the proof.
Similar to [22], we can draw the conclusion that the Poincaré series diverges at s = h, based on the estimation growth rate of VolS(x, r) in Thoerem C. This is the following corollary: The proof is the same as the proof of Corollary 5.2 in [22], so we omit it here.

Existence and uniqueness of the measure of maximal entropy
Let P : SX → X(∞) × X(∞) be the projection given by P (v) = (γ v (−∞), γ v (+∞)). Denote by I P = P (SX) = {P (v) | v ∈ SX} the subset of pairs in X(∞) × X(∞) which can be connected by a geodesic. Note that the connecting geodesic may not be unique. Fix a point p ∈ X, we can define a Γ-invariant measure µ on I P by the following formula (cf. [23] Lemma 2.4): dµ(ξ, η) = e h·βp(ξ,η) dµ p (ξ)dµ p (η), where β p (ξ, η) = −{b p (q, ξ) + b p (q, η)} is the Gromov product, and q is any point on a geodesic γ connecting ξ and η. It's easy to see that the function β p (ξ, η) does not depend on the choice of γ and q. In geometric language, the Gromov product β p (ξ, η) is the length of the part of the geodesic γ ξ,η between the horospheres H ξ (p) and H η (p). Then, µ induces a φ-invariant measure µ on SX with for all Borel sets A ⊂ SX. Here π : SX → X is the standard projection map and Vol is the induced volume form on π(P −1 (ξ, η)). By the definition of P , we know that if there is no geodesic connecting ξ and η, then P −1 (ξ, η) = ∅. If there are more than one geodesics connecting ξ and η, and one of them has rank k ≥ 1, then by the flat strip theorem, all of these connecting geodesics have rank k. Thus, P −1 (ξ, η) is exactly the k-flat submanifold connecting ξ and η, which is consisting of all the rank k geodesics connecting ξ and η. Especially, when k=1, P −1 (ξ, η) is exactly the rank 1 (thus unique) geodesic connecting ξ and η. From the above we can see the induced volume form Vol satisfies that for any Borel set A ⊂ SX and t ∈ R, Vol{π(P −1 (ξ, η) ∩ φ t A)} = Vol{π(P −1 (ξ, η) ∩ A)}. Therefore the Γ-invariance of µ leads to the Γ-invariance of µ.
By Theorem 3.15 we know the horospheres depend continuously on their centers at X(∞). While the value of the Gromov product is the length of the segment of the connecting geodesic between the horospheres, β p (·, ·) : X(∞) × X(∞) → R is a continuous function. Since X(∞) × X(∞) is compact (Proposition 2.4), β p (·, ·) is bounded function. By the fact that µ p is uniformly bounded and M is compact, µ can be projected to a finite φ-invariant measure on SM . To simplify the notation, we also use µ to denote this projected measure. If µ(SM ) = 1, we can normalize it. So in this paper we always assume µ is a probability measure. Now, we show that the measure µ on SM is the unique maximal entropy measure. Thus µ is independent on p. This measure is called the Knieper measure since it is first obtained by G. Knieper on rank 1 manifolds (cf. [23]). We decompose our proof into 3 parts: First, we show that h µ (φ) = h top (g), so µ is a measure of maximal entropy; Second, we show µ is an ergodic measure; In the last step, we show that the entropy of any invariant probability measure orthogonal to µ is strictly less than h µ (φ), this leads to the uniqueness of the invariant probability measure of maximal entropy. We should remark that this argument follows the idea of Knieper in [23].

Entropy of the Knieper measure
In this subsection, we will calculate the entropy of the Knieper measure µ and prove that where h = h top (g) is the topological entropy of the geodesic flow. We recall that the measure theoretical entropy h µ (φ) of the geodesic flow is defined to be the entropy of µ with respect to the time-1 map φ 1 , i.e. h µ (φ) := h µ (φ 1 ). Since by the variational principle, we know that h top (g) = sup where M(φ 1 ) denotes the set of all φ 1 -invariant measures. Then, (5.18) implies that µ is an invariant measure of maximal entropy.
For each k ∈ Z + , we define a metric d k on SM or SX defined by the following formula: φ is also a finite measurable partition of SM . For each v ∈ SM and α ∈ A Similar to Lemma 2.5 in [23], using Proposition 3.24(2) we can show that, on a compact rank 1 manifold without focal points, there is a > 0 such that for any α ∈ A n φ , we have µ(α) ≤ e −hn a. Therefore where h = h top (g) is the topological entropy of the geodesic flow. This implies that and then h µ (φ 1 ) = h. Thus we know that µ is an invariant measure of maximal entropy.

Ergodicity of the Knieper measure
In this subsection, we will prove that the Knieper measure µ is an ergodic measure of the geodesic flow. Our proof follows the idea of the proof in Section 4 of Knieper [23]. To simplify the notation, we also use µ to denote the Γ-invariant measure on SX defined in (5.17), which is exactly the lifting of the maximal entropy measure µ we are considering.
Lemma 5.28. Assume that v ∈ SX is a recurrent rank 1 vector, then for any w ∈ W s (v), where d 1 is the Knieper metric defined in Subsection 5.1.
Proof. First of all, since v is an recurrent vector on SX, then we can find a sequence {α n } ∞ n=1 ⊂ Γ and a sequence of time {t n } ∞ n=1 with t n → +∞, such that Now take an arbitrary vector w ∈ W s (v). In [36], R. Ruggiero showed that for manifold without focal points, γ v is positively asymptotic to γ w . Then by Lemma 2.1, the function is monotonely non-increasing. Now we show that d 1 (φ t (w), φ t (v)) → 0 as t → +∞, by contradiction. Assume that there is a constant c > 0 such that Then for each t n in (5.19), we write t = t n + s > 0 where s ∈ [−t n , +∞), then we have Then, Note that d 1 (φ tn (w), φ tn (v)) ≤ d 1 (v, w), and dα n (φ tn v) → v. So we know that for any ǫ > 0 there is a number N > 0 such that for all integer n > N , This implies that the set {dα n (φ tn w)} has an accumulate point. Without loss of generality, we assume dα n (φ tn w) → w ′ . Let n → +∞, we get that This means that the geodesics γ v and γ w ′ bound a flat strip, which contradicts the assumption that rank(v) = 1. We are done with the proof.
Let a be an arbitrary rank 1 axis, and U, V ⊂ X(∞) be a pair of neighborhoods of a(+∞) and a(−∞) satisfying Proposition 2.11. Therefore for each pair (ξ, η) ∈ U × V , there is a unique rank 1 geodesic connecting ξ and η.
Let f : SX −→ R is an arbitrary continuous Γ-invariant function. By Birkhoff ergodic theorem, for µ-almost all vectors v ∈ SX, f + (v) and f − (v) in the following are well-defined and equal: Where γ v is the unique geodesic with γ ′ v (0) = v. It is easy to see that f ± are constants along each geodesics. So we can define f ± (c) := f ± (c ′ (0)) for every geodesic c. Let Lemma 5.29. There is a geodesic c 1 ∈ G rec (U, V ) with c 1 (−∞) = ξ 1 ⊂ U , such that has full measure w.r.t. µ p , i.e., µ p (G ξ 1 ) = µ p (V ).
Take a ξ 1 ∈ U such that there is a geodesic c 1 ∈ G rec (U, V ) with c(−∞) = ξ 1 , and µ p (G ξ ) = µ p (V ). From the above argument we know that the point ξ 1 and the geodesic c 1 always exist. We are done with the proof.
The next lemma is a consequence of Lemma 5.28 and 5.29.
Proof. We follow the idea of the proof of Lemma 4.2 in [23]. Since we know that µ p (G ξ 1 ) = µ p (V ), then almost every c ∈ G rec (U, V ) satisfies that c(+∞) ∈ G ξ 1 . By the definition of G ξ 1 , we know that there is a geodesic c 2 ∈ G rec (U, V ) with c 2 (−∞) = c 1 (−∞) = ξ 1 and c 2 (+∞) = c(+∞). Then by Lemma 5.28, after a re-parametrization, we have Now, we are ready to prove the ergodicity of µ.
Theorem 5.31. The geodesic flow φ t is ergodic with respect to the measure µ.
Proof. It is sufficient to prove that for any continuous Γ-invariant function f : SX −→ R, the function f + is constant almost everywhere. Let By Lemma 5.28 and Lemma 5.30 we know f + (c) = f + (c 1 ) for almost all geodesics c with Then similar to our calculation in the proof of Lemma 5.29, we have Let Y = α∈Γ α( V ). Recall that U, V are neighborhoods of a(−∞) and a(+∞) respectively, where a is a rank 1 axis. Then by Lemma 2.12, one can see that µ p (Y ) = µ p (X(∞)). This implies that Z := {c ′ (t) | c(+∞) ∈ Y, t ∈ R} is a µ-full measure set in SX. Since f is a Γ-invariant function, f + is also Γ-invariant. Therefore, f + = f + (c 1 ) µ-a.e. on Z. This implies that f + is constant µ-almost everywhere on SX. So the geodesic flow φ t is ergodic with respect to the measure µ.

Uniqueness of the maximal entropy measure of geodesic flow
In this subsection, we will prove the measure µ is the unique maximal entropy measure, i.e. for any φ-invariant probability measure ν = µ, we have h ν (φ) < h µ (φ) = h top (g).
For any φ-invariant probability measure ν = µ, there is a unique decomposition where ν abs is an φ-invariant measure which is absolutely continuous with respect to µ, and ν sing is an φ-invariant measure which is singular with respect to µ.
This leads to the fact that for any t ∈ R, f •φ t (v) ≡ f (v), µ-a.e. v ∈ SM . Since µ is ergodic, we know f (v) = 1 for µ-a.e. v ∈ SM . Therefore ν abs = µ. Moreover, by the formula we know that, to prove the uniqueness of the maximal entropy measure, we just need to prove h ν (φ) < h top (g) for all φ-invariant measure ν which is singular with respect to µ. In the following, we will prove this inequality. We remark that our proof follows the idea of Knieper [23].
Take a compact fundamental domain F ⊂ X and a fixed reference point p ∈ F. Let R ′ > 2R + 1 2 be a constant such that F ⊂ B(p, R ′ ), where R > 0 is the constant defined in Proposition 3.24. Consider the unit tangent bundle SB(p, R ′ ) of B(p, R ′ ). Let Proposition 3.24 implies that for any x ∈ X with d(p, x) > R ′ + R there is a constant c ′ > 0 which is independent of x such that µ(D(x, R ′ , R)) ≥ c ′ e −hd(p,x) (cf. [23], Lemma 5.1). Then we have the following lemma: Lemma 5.32. Suppose x ∈ X satisfies d(p, x) ≥ R ′ + R + n, with n ∈ Z + . Then there is a constant a = a(δ, R ′ , R) > 0 such that the cardinality of any (d n , δ)-separated set of D(x, R ′ , R) is bounded by a.
Proof. Take a maximal δ 8 -separated set S 1 = {q 1 , q 2 , · · · , q k } of B(p, R ′ ), and a maximal Let c ij be the unique geodesic with c ij (0) = q i and c ij (d(q i , p j )) = p j .
For each L 2n i , i = 1, · · · , k(n), it is obvious that Since the set {v j ∈ V | B d 2n (v j , ǫ) ∩ L 2n i = ∅} can be lifted to a (d 2n , ǫ)-separated set of F 2n i ⊂ D(y i , R ′ , 2R) ⊂ SX, by Lemma 5.32 we know that ♯{v j ∈ V | B d 2n (v j , ǫ) ∩ L 2n i = ∅} ≤ a(2ǫ, R ′ , 2R). Therefore we get For each n ∈ Z + , Let A n = {φ n L 2n i | i = 1, · · · , k(n)}. Based on the definition of L 2n , it is easy to check that for each A ∈ A n and u, v ∈ A, there is a continuous curve l : [0, 1] → SM connecting u and v such that for all t ∈ [−n, n], we have length(π • φ t (l)) ≤ R ′ .
Following the discussion in Theorem 5.31, we know that the regular set in SM has positive µ-measure. Then by the ergodicity of geodesic flow with respect to µ, we get µ(sing) = 0. The following technical lemma is Lemma 5.5 of [23]. See also Lemma 9.6 of [24]. It holds because we still have the flat strip lemma in no focal point case.
Lemma 5.33. Let ν be a Borel probability measure on SM . For a fixed constant b > 0, let A n be a sequence of measurable coverings of SM such that for each n and v, w ∈ A ∈ A n there exists a continuous curve α : [0, 1] → SM with α(0) = v, α(1) = w, and Lg(πφ t (α)) ≤ b for all t ∈ [−n, n]. Let Ω = SM be a set containing all singular vectors. Then there exists a union C n of subsets of A n such that ν(Ω △ C n ) := ν(C n \ Ω) + ν(Ω \ C n ) → 0.
Furthermore, we can show that for any φ-invariant measure ν which is singular with respect to µ. there is subset C n ⊂ SM which is the union of some subsets of A n such that Now we consider the entropy of ν mentioned above, i.e., ν is a φ-invariant measure which is singular with respect to µ. Since the geodesic flows on the manifolds without focal points are entropy-expansive (cf. [26], and we will discuss the entropy-expansiveness in the next section), and diam d 2n (B n ) ≤ 2ǫ = 1 3 Inj(M ), by Theorem 3.5 of [7] and Corollary 3.4 of [23], we know that We divide B n into two subsets: Let a n = B∈B n + ν(B). Obviously 1 > a n ≥ ν(C n ) → 1 as n → ∞. Then we have: where the second inequality comes from Lemma 5.7 in [23]. By the same calculation as the one in [23], we have the following estimations: Thus where K = α β a(ǫ, R ′ , 2R) is independent of n.
Since 1 ≥ a n ≥ ν(C n ) → 1 and µ(C n ) → 0, we know the righthand side of the above inequality approaches to −∞ as n → ∞. Therefore we must have h ν (φ 1 ) < h = h top (g).
We are done with the proof of the uniqueness of the maximal entropy measure for the geodesic flow on the manifold without focal points.
6 Entropy gap and growth rate of singular and regular closed geodesics In this section, we are going to discuss the distribution of the primitive closed geodesics on compact rank 1 manifolds without focal points. First, we will prove the entropy gap h top (φ| sing ) < h top (g), which is the second part of Theorem A. Based on this result, We can show that the the ratio P sing (t) Preg(t) decays exponentially as t → +∞. This means that the set of the regular primitive closed geodesics grows exponentially faster than the set of singular primitive closed geodesic. To show this, we need to consider the upper semi-continuity of h µ (φ) : M inv (φ) → R for the geodesic flow.
We use φ 1 to denote by the time-1 map of the geodesic flow. For each v ∈ SM and ǫ > 0, define where d 1 is the Knieper metric defined in Subsection 5.1. In [7], Bowen showed that if T : X → X is entropy-expansive, then the map µ → h µ (T ) is upper semi-continuous. It was showed in [26] that if the compact manifold (M, g) is bounded asymptote and have no conjugate points, then the geodesic flow on SM in entropyexpansive. Since a manifold without focal points is always bounded asymptotic (cf. [37]), the geodesic flow on it is entropy-expansive. Combining the two results in the above, we have the conclusion that the map µ → h µ (φ) is upper semi-continuous. This is the following proposition. Based on the discussion in the above, we can prove the second part of Theorem A. Proof. We prove this theorem by contradiction. Assume that h top (φ| sing ) = h top (g). Then, since the singular set sing is a closed φ-invariant subset of SM , we can find a sequence of invariant probability measures ω k supported on sing with h ω k (φ) ≥ h top (φ| sing ) − 1 k .
By the compactness of M inv , the set {ω k } k∈Z + has an accumulate point ω (under the weak * topology). Obviously supp(ω) ⊂ sing.
This contradicts to the uniqueness of the invariant measure of maximal entropy. We have proved that h top (φ| sing ) < h top (g).
Finally, we are ready to proof Theorem D which is restated below.
Theorem D. Let (M, g) be a compact rank 1 Riemannian manifold without focal points. Then there exist a > 0 and t 1 > 0 such that for all t > t 1 , e ht at ≤ P reg (t) ≤ ae ht .
Moreover, there exist positive constants ǫ and t 2 such that P sing (t) P reg (t) ≤ e −ǫt , t > t 2 .
Proof. It was shown in [22] that, for the geodesic flow defined on a compact rank 1 manifold with nonpositive curvature, there is a t 1 > 0 such that for all t > t 1 , e ht at ≤ P reg (t) ≤ ae ht , (6.21) for some a > 0. One can check that this proof is also valid for manifolds without focal points. In fact, the upper bound for P reg (t) is a direct consequence of Theorem C. The lower bound was proved for rank 1 manifolds with nonpositive curvatures by Knieper in [22], Theorem 5.8. This theorem relies on Lemma 5.6 and 5.7 in [22], which was originally proved by Knieper in [21]. We will exhibits the no focal points version of these two lemmas. And then the lower bound follows. Here we define the function l(α) := min{d(p, α(p)) | p ∈ X}, for any α ∈ Γ.
This lemma corresponds to Lemma 5.6 in [22].
Similar as Lemma 6.37, this lemma is also a consequence of our results in Section 2 (Proposition 2.11 and 2.14). Because of the limitation of the length of this article, we omit the proof at here. Readers can give the proof by follow Knieper method of in [21] without any difficulty.
To prove the second part, we will follow the idea in the proof of Corollary 6.2 in [23]. Suppose h top (g) = h. Let ǫ 0 = h−htop(φ| sing ) 2 . First we consider the numerator P sing (t) = ♯P sing (t). For each t > 0, let n(t) = [t] + 1 where [t] is the largest integer less or qual to t. Let δ = Inj(M ) which is the injective radius of M . Then it is easy to see that all the primitive closed geodesics in P sing (t) form a (t(n), δ)-separated set in sing. This leads to the inequality: lim sup t→+∞ 1 t log P sing (t) ≤ h top (φ| sing ) < h top (g) − ǫ 0 = h − ǫ 0 . (6.22) Now consider denominator P reg (t) = ♯P reg (t). We already explained that the formula (6.21) holds for geodesic flows on compact manifolds without focal points. Combining the two inequalities (6.22) and (6.21) in the above, take ǫ < ǫ 0 , then there is a t 2 > 0 such that for all t > t 2 , P sing (t) P reg (t) ≤ e −ǫt .
We are done with the proof.
Fund from China Scholarship Council (CSC). W. Wu is partially supported by NSFC under Grant Nos.11701559 and 11571387. All of us would like to express our great gratitude to the anonymous referee for her/his tremendous suggestions and comments on the first version of the article.