Ergodic properties of some negatively curved manifolds with infinite measure
Abstract.
Let be a geometrically finite negatively curved manifold with fundamental group acting on by isometries. The purpose of this paper is to study the mixing property of the geodesic flow on , the asymptotic equivalent as of the number of closed geodesics on of length less than and of the orbital counting function .
These properties are well known when the BowenMargulis measure on is finite. We consider here divergent Schottky groups whose BowenMargulis measure is infinite and ergodic, and we precise these ergodic properties using a suitable symbolic coding.
Contents
Acknowledgements. This paper comes from my PhD at the University of Nantes between 2013 and 2016 under the direction of Marc Peigné and Samuel Tapie. During that time, my research was supported by Centre Henri Lebesgue, in programm “Investissements d’avenir”  ANR11 LABX002001. I want to thank Sébastien Gouëzel for his help and all his explanations during the writing of this paper.
1. Introduction
1.1. Background and previous results
Let be a connected, simply connected and complete riemannian manifold with pinched negative sectional curvature. Denote by the distance on induced by the riemannian structure of and by a discrete group of isometries of , acting properly discontinuously without fixed point and let . Fix . The study of quantities like the orbital function
is strongly related to the one of the dynamic of the geodesic flow on the unit tangent bundle of the quotient manifold. Let us first define precisely this flow: each couple determines a unique geodesic satisfying and for any , the action of is given by . It is known (see [33]) that the topological entropy of the geodesic flow is given by the rate of exponential growth of the orbital function, that is
This last quantity is also the critical exponent of the Poincaré series of the group defined as follows: for any
S. J. Patterson (in [35]) and D. Sullivan (in [40]) used these series to construct a family of measures , the socalled PattersonSullivan measures. More precisely, each measure is fullysupported by the limit set , which is defined as the set of all accumulation points of one(all) orbit(s) in the visual boundary of . This set also is the smallest nonempty invariant closed subset of . It is the closure in the boundary of the set of fixed points of . A group is said to be elementary if its limit set is a finite set. S.J. Patterson and D. Sullivan described a process to associate to this family a measure defined on , which is invariant under the action of the geodesic flow. When the group is divergent, i.e. (otherwise is said to be convergent), the family is unique up to a normalization, hence is also unique; this is the case when has finite mass. We will focus in this paper on the case of divergent groups, which allows us to speak about “the” BowenMargulis measure even when it has infinite mass. Nevertheless, in this introduction, the assumption “ has infinite BowenMargulis measure“ should be in general understood as the fact that any invariant measure obtained from a PattersonSullivan density has infinite mass. We first study here a property of mixing of the geodesic flow with respect to this measure. We say that the geodesic flow is mixing with respect to a measure with finite total mass on , if for any measurable sets , one gets
(1) 
When the measure has infinite mass, this definition may be extended saying that the flow is mixing if
When the measure is finite, Property (1) was first proved in [24] for finite volume surfaces in constant curvature, by F. Dal’bo and M. Peigné for Schottky groups with parabolic isometries acting on Hadamard manifolds with pinched negative curvature (see [13]) and by M. Babillot in [1]. The following result of T. Roblin [37] gathers all the information known in such a general content.
Theorem (Roblin).
If the BowenMargulis measure has finite mass (resp. infinite mass), the flow satisfies
Remark 1.1.
The definition of mixing in infinite measure seems to be weak (see the third chapter of [37] about this fact). Nevertheless, our Theorem A below will furnish an asymptotic of the form
which can be understood as a mixing property, up to a renormalization.
On the one hand, this property is interesting from the point of view of the ergodic theory. On the other hand, in the case of geometrically finite manifolds with finite measure, the property of mixing of the geodesic flow may be used to find an asymptotic of the orbital function . This idea was initially developped in G.A. Margulis’ thesis [31] for compact manifold with negative curvature: the mixing of the geodesic flow implies a property of equidistribution of spheres on , which leads to the orbital counting. In the constant curvature case, other proofs of the orbital counting have been developped using spectral theory (see for instance [30]) or symbolic coding (see [29]). In [37], the author generalizes the ideas of Margulis and deduces the asymptotic for the orbital counting function from the mixing of the geodesic flow. He shows the following
Theorem (Roblin).
Let be a complete manifold with pinched negative sectional curvatures. If the BowenMargulis measure has finite mass (resp. infinite mass), the asymptotic behaviour of the orbital function is given by
where is the mass of the PattersonSullivan measure .
We eventually focus on finding an asymptotic equivalent for the number of closed geodesics on of length less than , as goes to infinity. Such asymptotic was first found out by Selberg for compact hyperbolic surfaces (see [39]), then by Margulis ([31]) for compact manifold of negative variable curvature and extended in [34] to periodic orbits of axiomA flows. This was generalized by T. Roblin in [37] as follows.
Theorem ([37]).
Let be a geometrically finite complete manifold with sectional curvatures less than , whose BowenMargulis measure has finite mass. For all , let be the number of closed geodesics on of length less than . Then as goes to infinity,
When the BowenMargulis measure is infinite, Roblin’s method does not yield to such asymptotic. We will detail why in the next paragraph.
1.2. Assumptions and results
In this article, we will focus on some manifolds , where is a divergent group whose BowenMargulis measure on has infinite mass and whose Poincaré series are controlled at infinity. For such manifolds, we establish a speed of convergence to of the quantities for any with finite measure, an asymptotic equivalent for the orbital counting function and an asymptotic lower bound for the number of closed geodesics . The groups which we consider are exotic Schottky groups, whose construction was explained in the articles [10] and [36] and will be recalled in the second section. The main idea of these papers is the following: let be a geometrically finite hyperbolic manifold with dimension , with a cusp and whose fundamental group is a non elementary Schottky group. Theorems A and B in [10] ensure that the group is of divergent type and that is finite. The proofs of these results give a way to modify the metric in the cusp in order to obtain a manifold isometric to a quotient where

the manifold is a Hadamard manifold with pinched negative curvature;

the group acts by isometries on and is of convergent type; the measure is thus infinite.
The article [36] extends the previous construction and allows us to modify the metric in the cusp of in such a way that the group is of divergent type with respect to the metric on and the measure is still infinite. This article furnishes examples of manifolds on which our work applies.
Let be a Hadamard manifold with pinched negative curvature between and , where and let be a Schottky group, i.e. is generated by elementary groups in Schottky position (see Paragraph 2.1.2), where and . Assume that for some , the groups satisfy the following family of assumptions :

The group is of divergent type.

For any , the group is parabolic, of convergent type and its critical exponent is equal to .

There exists a slowly varying function ^{1}^{1}1A function is slowly varying at infinity if for any , it satisfies . such that for any , the tail of the Poincaré series at of the group satisfies
for some constant .

For any , the group satisfies the following property
We make additionally the following assumption:

For any , there exists such that for any and any large enough
We will say that the parabolic groups are ”influent“, since their properties will determine all the dynamical properties of on . The reader should notice that each of these subgroups is convergent and has the same critical exponent as the whole group . The existence of at least a factor having these properties is needed to get a Schottky group with infinite BowenMargulis measure, see [10] and Section 2 below. On the contrary, the groups are said to be “noninfluent” and their own critical exponent may be in particular strictly less than .
By a theorem of Hopf, Tsuji and Sullivan, the divergence of ensures that the geodesic flow is totally conservative with respect to the measure . Under these assumptions, we first show the following theorem, which precises the rate of mixing of the geodesic flow.
Theorem A.
Let be a Schottky group satisfying Hypotheses for some and be two measurable sets with finite measure.

If , there exists a constant such that

if , there exists a constant such that
where for any .
The proof of this result relies on the study of a coding of the limit set of and on a symbolic representation of the geodesic flow given in the fourth section.
In Section 7, we establish an asymptotic lower bound for the number of closed geodesics with length less than . We prove
Theorem B.
Let be a manifold with pinched negative curvature whose fundamental group satisfies Hypotheses for . Then
We could not improve our proof to get a full asymptotic equivalent for . This lower bound remains nevertheless surprising. Let us recall the following result proved in [37]. For all , let be the set of closed orbits for the geodesic flow on with period less than . For any closed orbit , let be the normalized Lebesgue measure along . Then as ,
(2) 
in the dual of the set of continuous functions with compact support in . In particular, when is convexcocompact ( is thus finite in this case), the set is compact and (2) applied with implies the counting result . When is geometrically finite with finite BowenMargulis measure, Roblin shows that (2) still implies . When the BowenMargulis measure has infinite mass, Roblin shows that, still in the dual of compactly supported continuous functions of ,
Therefore, we could have expected that the would have been negligible with respect to as goes to ; Theorem B above thus contradicts this intuition.
We eventually establish the following asymptotic equivalent for the orbital counting function.
Theorem C.
Let be a Schottky group satisfying the family of assumptions for some .

If , there exists such that

If , there exists such that
where for any .
To prove Theorem C, we need to extend the coding of the points of the limit set to the orbit of some point in the boundary at infinity.
The constants appearing in Theorems A and C will be precised in the proofs.
Remark 1.2.
In his seminal work, T. Roblin always assumes the nonarithmeticity of the length spectrum, i.e. the set of lengths of closed geodesics in is not contained in a discrete subgroup of . This assumption is satisfied in our setting, because the quotient manifold has cusps (see [12]).
Remark 1.3.
In the case of a Schottky group with only two factors, satisfying hypotheses for some and with at least one influent factor, the results presented above are still valid but their proofs are slightly more technical. Indeed, the transfer operator will then have two dominant eigenvalues and (see [3] and [12]). The proof of our result hence would have to be adapted in this case, similarly to the arguments in [12], which we will not do here.
Let us now explain why we present separately the additional assumption in the family . In [15], the authors prove a result similar to Theorem C for , without this assumption. But they can not obtain it for , their proof (at section 6) being based on the renewal theorem of [18], which does not ensure any more a true limit when . Our arguments rely on the article [21] and we avoid this distinction between and thanks to the additional assumption . We do not know whether this hypothesis is a consequence of the first four in our geometric setting.
Remark 1.4.
We may notice that the assumption is equivalent to each of both following statements:

For any , there exists a constant such that for any and any large enough, the following inequality is satisfied

There exists a constant such that for any and any large enough
The equivalence between the statements and is clear and the fact that implies follows from definitions. We may find a proof of the equivalence between the statements and in Proposition 2.5 of [14] (in the finite volume case). In section 3, we detail another proof of the converse property involving Karamata and Potter’s lemmas.
Remark 1.5.
In the family of hypotheses for , the “influent” parabolic groups are supposed to be elementary. Their rank may be larger than . Nevertheless, in the proofs of Proposition 4.26 and 6.4 and of Facts 8.3 and 8.4, we work with parabolic groups of rank in order to simplify the notations. The arguments are always true in higher rank.
1.3. Outline of the paper
The second section is devoted to the construction of Schottky groups satisfying hypotheses as explained in [10] and [36].
The third section is devoted to the presentation of some properties of stable laws with paramater for random variables, together with results on regularly varying functions. These will be crucial tools in the sequel due to our assumptions and .
In the fourth section, we define a coding of the limit set of and of the geodesic flow. We then introduce a family of transfer operators associated to this coding. We finally end this part with a study of the spectrum and of the regularity of the spectral radius of these operators.
The proof of the mixing rate given in Theorem A in the case is exposed in Section 5: it is based on the one of Theorem 1.4 in [21]. The case is presented in Section 6 and the approach is inspired by [32].
We establish in Section 7 the asymptotic lower bound for closed geodesics given in Theorem B.
In section 8, we extend the previous coding of the limit set to include the orbit of a base point and study the family of extended transfer operators associated to this new coding. These extended operators will be central in the proof of Theorem C.
The final section is dedicated to the proof of Theorem C, which follows the same steps as Theorem A.
Notations:
In this paper, we will use the following notations. For two functions , we will write (or ) if for a constant and large enough and (or ) if and . Similarly, for any real numbers and and , the notation means .
If are two subsets of , we denote by .
The value of the constant which appears in the proofs may change from line to line.
2. About “exotic” Schottky groups
In this section, we first recall some definitions and properties about manifolds in negative curvature. Then we give a sketch of the construction of exotic Schottky groups following [10] and [36].
2.1. Negatively curved manifolds and Schottky groups
2.1.1. Notations
Let be a Hadamard manifold of pinched negative curvature with , endowed with the distance induced by the metric. We denote by its boundary at infinity (see [4]). For a point and two points , the Busemann cocycle is defined as the limit of when goes to . This quantity represents the algebraic distance between the horospheres centered at and passing through and respectively. This function satisfies the following property : for any and
(3) 
The Gromov product of two points and of seen from the point is given by the following formula
where is any point on the geodesic with endpoints and ; this product does not depend on the point . The curvature being bounded from above by , we may find in [8] a proof of the fact that the quantity defines a distance on , which satisfies the following “visibility” property: there exists a constant depending only on the bounds of the curvature of such that for any
(4) 
As a consequence of this property, we mention the following important lemma.
Lemma 2.1 (triangular “quasiequality”).
Let such that . There exists a constant such that for any and .
This result can be proved for instance using the arguments given in section 2.3 in [38]. This lemma thus furnishes a complement to the classical triangular inequality; many results of this paper are based on it. Denote by the group of orientationpreserving isometries of . The action of can be extended to by homeomorphism. It follows from the previous definitions that for any and any
(5) 
We thus talk about “conformal action” of on the boundary at infinity; the conformal factor of an isometry at the point is given by the formula . From (3), we deduce that the function satisfies the following cocycle relation: for any and any
2.1.2. Schottky product groups
Let and be isometries satisfying the following property: there exist nonempty pairwise disjoint closed subsets of such that for all and all , one gets . The isometries are said to be in Schottky position. Klein’s pingpong lemma implies that the group generated by the isometries is free and acts properly discontinuously and without fixed point on . The group is called a Schottky group. The limit set of such a group is a perfect nowhere dense set.
Let us give an example in the model of the Poincaré halfplan. Let be a parabolic isometry; the nontrivial powers of send into , hence we set . On the other hand, let us conjugate the hyperbolic isometry by . The soobtained isometry is hyperbolic with fixed points and ; its negative powers send into , and its positives send into . We finally set (see figure 1 below).
We extend this definition for product of groups. Fix . We say that groups are in Schottky position if there exist nonempty pairwise disjoint closed subsets of such that for all , one gets . We may note that contains the limit set of . Thus the group generated by is the free product ; it is called Schottky product of the groups . Each group , , is called a Schottky factor of .
In the sequel, we will need to consider subsets of with the same dynamical properties as the sets under the action of . For any , we introduce sets , which are geodesically convex and connected (resp. admits two geodesically convex connected components) when is generated by a parabolic (resp. hyperbolic) isometry and whose intersection with contains ; in addition, we assume that and that these sets are pairwise disjoint. Figure 2 illustrates the situation for the above isometries and .
2.1.3. Geodesic flow and BowenMargulis measure
Using Hopf coordinates, we identify the unit tangent bundle to the set : the point determines a unique triplet in , where and are the endpoints of the oriented geodesic passing through at time with tangent vector and . The group acts on as follows: for any
and the action of the geodesic flow is given by
for any . These two actions commute and, quotienting by , define the action of the geodesic flow on . By [17], the nonwandering set of on is .
By Patterson’s construction (see [35] and [40]), there exists a family of finite measures on supported on and satisfying, for any , any and :
(6) 
where for any Borel subset of and is the Poincaré exponent of . As soon as is divergent and geometrically finite, the measures do not have atomic part (see [10]). As observed by Sullivan [40], the Patterson measure of may be used to construct an invariant measure for the geodesic flow with support . It follows from (6) and (5) that the measure defined on by
is invariant so that on is both and invariant. It thus induces on an invariant measure for . When this measure has finite total mass, this is the unique measure which maximises the measuretheoretic entropy of the geodesic flow restricted to its non wandering set: it is called the BowenMargulis measure (see [33]). When this measure is infinite, there is no finite invariant measure which maximises the entropy: however, we still call it the BowenMargulis measure.
2.2. Construction of exotic Schottky groups
Let us recall the genesis of the setting in which we will work, i.e. the construction of some exotic Schottky groups introduced in [10] and [36].
2.2.1. Divergents groups and finite BowenMargulis measure
The article [10] gives the first known example of geometrically finite manifolds with pinched negative curvature and whose BowenMargulis measure has infinite mass.
The authors construct such examples by providing convergent Schottky groups (which hence are geometrically finite). Therefore, the BowenMargulis measure on has infinite mass.
To get such examples, the authors first show that the group needs to contain a parabolic subgroup whose critical exponent is .
Theorem A ([10]).
Let be a geometrically finite group with parabolic transformations. If for any parabolic subgroup , then is of divergent type.
The assumption for any parabolic subgroup is called the “critical gap property” of the group . It follows from Proposition 2 of [10]: if is a Schottky group, whose each factor , , are divergent, then it has the critical gap property.
When the group is divergent, but still has parabolic elements, necessary and sufficient condition for the finiteness of the BowenMargulis measure is given by the following criterion.
Theorem B ([10]).
Let be a divergent geometrically finite group containing parabolic isometries. The measure is finite if and only if for any parabolic subgroups of , the series converges.
We can deduce at least two things from both previous theorems. On the one hand, a geometrically finite group containing parabolic isometries satisfying the critical gap property is divergent and admits a finite measure . On the other hand, we understand that a first step to obtain a group with infinite measure involves the construction of parabolic groups of convergent type. This is the purpose of the next paragraph, which is based on [10].
2.2.2. Construction of convergent parabolic groups
Let us first consider the situation in constant curvature . Fix . We may identify with the product endowed with the metric . Let an elementary parabolic group acting on . Up to a conjugacy, we may suppose that the elements of fix the point at infinity . Denote by the horoball centered at and passing through . The group acts by euclidean isometries on the horosphere . By a Bieberbach’s theorem (see [5] and [6]), there exists a finite index abelian subgroup of which acts by translations on a subspace , . There thus exist linearly independant vectors and a finite set such that any element decomposes into for and , where is the th power of the translation of vector . In this case, the Poincaré series of is given by: for
The quantity
is bounded when , where is the euclidean norm in . The previous series thus behaves like the following
which diverges at its critical exponent .
In the sequel, following [10], we will modify the metric in the horoball in such a way that the parabolic group will still have critical exponent , but its Poincaré series will converge at . In this purpose, we consider another model of the hyperbolic space, which will be more suitable to understand the action of on the horospheres. The classical upper half space model of the hyperbolic space, is isometric to via the diffeomorphism
Let us denote by the horoball of level centered at infinity in this model; one gets . Fix and let us denote and for ; these two points both belong to the horosphere , and the distance between them, with respect to the metric on induced by the hyperbolic metric on , is equal to , where is the Euclidean norm on . Therefore, on the horosphere of level , the distance induced on the horosphere between and is . Since the curve is a quasigeodesic, we can deduce from [25] that the quantity is bounded. Let us now consider on the metric , where is chosen such that has pinched negative curvatures. Let us write for the distance induced by on . The same argument as previously given for the hyperbolic space shows that if , , then is bounded uniformly in , where is defined by the implicit equation for all . When and , we obtain the previous model of the hyperbolic space. One of the steps in [10] section 3 and [36] section 2 is to explain how the functions and have to be chosen so that the sectional curvature remains negative and pinched on endowed with . More precisely, Lemma 2.2 in [36] states the following.
Lemma 2.2.
Fix a constant . For any , there exist and a nondecreasing function satisfying:

if ;

if ;

if for any and , then ;

for any and the derivatives of tend to as goes to , uniformly in .
We may notice that this metric coïncides with the hyperbolic one on the set ; we can enlarge this area shifting the metric along the axis (see Paragraph 2.2 in [36] and Paragraph 2.2.4 of this paper).
On , the group defined above still acts by isometries and its Poincaré series behaves like
This series still admits as critical exponent and is convergent if and only if . In the next paragraph, we will see how to adapt the above construction of metric , , to highlight the existence of convergent parabolic group satisfying the assumptions and
2.2.3. On convergent parabolic group satisfying assumptions and .
Here we fix ,but the following construction may be adapted in higher dimension. Let be the point in and the translation of vector . As mentionned previously, for these metrics , , there exists such that for sufficiently large, one gets
As we saw in paragraph 2.2.2, this is enough to ensure the convergence of the parabolic group . Nevertheless, this estimate is not precise enough to ensure that satisfies Hypotheses and . Therefore, in the sequel, we present new metrics , , close to those presented in Lemma 2.2, for which we can precise the behaviour of the bounded term as .
Let us fix . For all real greater than some to be chosen later, let us set
where is a slowly varying function on with values in . Without loss of generality, we assume that is on and its derivates , satisfy as ([7], Theorem 1.3.3); furthermore, for any , there exist and such that for any
(7) 
Notice that