Ergodic properties of some negatively curved manifolds with infinite measure
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 Bowen-Margulis measure on is finite. We consider here divergent Schottky groups whose Bowen-Margulis measure is infinite and ergodic, and we precise these ergodic properties using a suitable symbolic coding.
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” - ANR-11- LABX-0020-01. I want to thank Sébastien Gouëzel for his help and all his explanations during the writing of this paper.
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 ) 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 ) and D. Sullivan (in ) used these series to construct a family of measures , the so-called Patterson-Sullivan measures. More precisely, each measure is fully-supported 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 non-empty -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” Bowen-Margulis measure even when it has infinite mass. Nevertheless, in this introduction, the assumption “ has infinite Bowen-Margulis measure“ should be in general understood as the fact that any invariant measure obtained from a Patterson-Sullivan 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
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  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 ) and by M. Babillot in . The following result of T. Roblin  gathers all the information known in such a general content.
If the Bowen-Margulis measure has finite mass (resp. infinite mass), the flow satisfies
The definition of mixing in infinite measure seems to be weak (see the third chapter of  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  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 ) or symbolic coding (see ). In , 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
Let be a complete manifold with pinched negative sectional curvatures. If the Bowen-Margulis measure has finite mass (resp. infinite mass), the asymptotic behaviour of the orbital function is given by
where is the mass of the Patterson-Sullivan 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 ), then by Margulis () for compact manifold of negative variable curvature and extended in  to periodic orbits of axiom-A flows. This was generalized by T. Roblin in  as follows.
Let be a geometrically finite complete manifold with sectional curvatures less than , whose Bowen-Margulis 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 Bowen-Margulis 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 Bowen-Margulis 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  and  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  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  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 111A 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 Bowen-Margulis measure, see  and Section 2 below. On the contrary, the groups are said to be “non-influent” 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.
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
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 . 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 ,
in the dual of the set of continuous functions with compact support in . In particular, when is convex-cocompact ( is thus finite in this case), the set is compact and (2) applied with implies the counting result . When is geometrically finite with finite Bowen-Margulis measure, Roblin shows that (2) still implies . When the Bowen-Margulis 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.
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.
In his seminal work, T. Roblin always assumes the non-arithmeticity 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 ).
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  and ). The proof of our result hence would have to be adapted in this case, similarly to the arguments in , which we will not do here.
Let us now explain why we present separately the additional assumption in the family . In , 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 , which does not ensure any more a true limit when . Our arguments rely on the article  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.
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  (in the finite volume case). In section 3, we detail another proof of the converse property involving Karamata and Potter’s lemmas.
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 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 . The case is presented in Section 6 and the approach is inspired by .
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.
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
2.1. Negatively curved manifolds and Schottky groups
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 ). 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
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  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
As a consequence of this property, we mention the following important lemma.
Lemma 2.1 (triangular “quasi-equality”).
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 . 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 orientation-preserving isometries of . The action of can be extended to by homeomorphism. It follows from the previous definitions that for any and any
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 non-empty pairwise disjoint closed subsets of such that for all and all , one gets . The isometries are said to be in Schottky position. Klein’s ping-pong 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 half-plan. Let be a parabolic isometry; the non-trivial powers of send into , hence we set . On the other hand, let us conjugate the hyperbolic isometry by . The so-obtained 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 non-empty 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 Bowen-Margulis 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 , the non-wandering set of on is .
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 ). As observed by Sullivan , 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 measure-theoretic entropy of the geodesic flow restricted to its non wandering set: it is called the Bowen-Margulis measure (see ). When this measure is infinite, there is no finite invariant measure which maximises the entropy: however, we still call it the Bowen-Margulis measure.
2.2. Construction of exotic Schottky groups
2.2.1. Divergents groups and finite Bowen-Margulis measure
The article  gives the first known example of geometrically finite manifolds with pinched negative curvature and whose Bowen-Margulis measure has infinite mass.
The authors construct such examples by providing convergent Schottky groups (which hence are geometrically finite). Therefore, the Bowen-Margulis 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 ().
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 : 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 Bowen-Margulis measure is given by the following criterion.
Theorem B ().
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 .
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  and ), 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
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 , 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 quasi-geodesic, we can deduce from  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  section 3 and  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  states the following.
Fix a constant . For any , there exist and a non-decreasing function satisfying:
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  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 (, Theorem 1.3.3); furthermore, for any , there exist and such that for any