G Gromykov

M Ould Haye

A Philippe
email: anne.philippe@univ-nantes.fr

A Frequency-domain test for long range dependence

Keywords: Long memory, dependence, time series, limit theorem, hypothesis test

A new frequency-domain test statistic is introduced to test for short memory versus long memory. We provide its asymptotic distribution under the null hypothesis and show that it is consistent under any long memory alternative. Some simulation studies show that this test is more robust than various standard tests in terms of empirical size when the normality of observed process is lost.

Introduction

Let us consider a stationary process (X t ) with a spectral density of some semi-parametric form:

f (λ) = |λ| -2d g(λ) (1)
with 0 ≤ d < 1/2 and g is an even, positive, continuous function on [-π, π]. We say that the process (X t ) is of short memory if d = 0 (i.e. the spectral density f is continuous over [-π, π]), and we say that the process is of long memory if 0 < d < 1/2 (i.e. the spectral density is unbounded at zero).

In time-domain terms, short memory means that the covariance function is summable (i.e. covariance function γ(h) = Cov(X i , X i+h ) goes to zero fast as the lag h increases) and long memory means that the covariance function is not summable (i.e. the covariance function goes to zero slowly as the lag increases). Literature is full of real world examples of both types of processes. Under mild conditions, the switch between time-domain and frequency-domain is easily made via the two representations Since the so-called re-scaled range statistic R/S was introduced by Hurst (1951), many authors have come with several time-domain statistics for testing long memory, such as modified R/S, KPSS (Kwiatkowsk-Phillips-Schmidt-Shin), rescaled variance statistic V/S (See [START_REF] Mandelbrot | Computer experiments with fractional Gaussian noises, Parts 1, 2, and 3[END_REF], [START_REF] Kwiatkowski | Testing the null hypothesis of stationarity against the alternative of a unit root[END_REF], [START_REF] Lee | On the power of the KPSS test of stationarity against fractionallyintegrated alternatives[END_REF], [START_REF] Giraitis | Rescaled variance and related tests for long memory in volatility and levels[END_REF], [START_REF] Giraitis | A test for stationarity versus trends and unit roots for a wide class of dependent errors[END_REF]). See [START_REF] Beran | Long-memory processes[END_REF] for a full description of all of these statistics. According to the simulation results provided in [START_REF] Giraitis | Rescaled variance and related tests for long memory in volatility and levels[END_REF], V/S statistic produces testing procedure with better empirical size and power as compared to the other time domain statistics. A detailed description of V/S statistic is given in Section 3. One common drawback in these statistics is the difficulty to calibrate the window parameters in order to ensure the right empirical size ( in particular for the small sample) which comes with the heavy price of a very weak power. Also, evaluation of their limiting distributions often involves approximating stochastic integrals. Time-domain based tests often suffer from the so-called high empirical size (i.e. the ratio of wrongly rejecting the null hypothesis is much bigger than the nominal level α). Here we address this problem by introducing a very promising frequency-domain statistic, for testing long memory versus short memory. This turns out to be very convenient and yields very simple limiting distributions. Our test statistic can be viewed as a contribution to this approach initiated by Lobato and Robinson (1998 (see Section 3 for a detailed description) and more recently by [START_REF] Bailey | Spectral approach to parameter-free unit root testing[END_REF] in testing unit root. However, our method bears many differences as will be explained later. The main interest of our approach is to preserve a good empirical size even when the underlying distribution is not Gaussian. From a sample X 1 , . . . , X n of the process (X t ), we are interested in building a testing procedure to discriminate between short and long range dependence. The proposed statistic is inspired from the well-known fact (see for example Giraitis et al. Theorem 5.3.1 (2012) or Moulines and Soulier 2003, section 3.1) that, for a large class of weakly dependent stationary processes, at any Fourier frequency, the normalized periodogram is asymptotically exponentially distributed. That is, for any fixed j, with λ j = 2πj/n,

f (λ) = 1 2π ∞ h=-∞ e -ihλ γ(λ) = 1 2π γ(0) + 2
I n (λ j ) f (λ j ) d -→ E, as n → ∞ (2)
where E has exponential distribution with mean 1, and where I n is the periodogram

I n (λ) = 1 2πn n j=1 X j e ijλ 2 .
Moreover, for different Fourier frequencies, the variables on the left hand side of (2) are asymptotically independent (see same references above). Since under short memory, the periodogram is an asymptotically unbiased estimator of the spectral density f , we estimate f (λ j ) by a sample average of periodograms. The procedure is as follows: we split our initial sample X 1 , . . . , X n into m blocks (or epochs), each of size , and we construct the periodogram I n,i on the ith block X (i-1) +1 , . . . , X i . The construction of this averaging of periodograms is known as Bartlett method to estimate the spectral density. Then we define our test statistic as follows:

Q n,m (s) = s j=1 I n (λ j ) 1 m m i=1 I n,i (λ j ) . ( 3
)
where s is the number of Fourier frequencies we want to include in the test.

We assume that m = m(n), and = (n) → ∞ as n → ∞. It is very important to emphasize the fact that m and increase with n and are not constant, and that m = n/ . We are simply using notation m and rather than m(n) and (n) only for the sake of simplicity. We should mention that our context is different from that of [START_REF] Reisen | On the properties of the periodogram of a stationary long-memory process over different epochs with applications[END_REF], where the number of epochs is fixed.

Main results

In this section we give the two asymptotic distributions of Q n,m (s) under (a) the null hypothesis "the process has a short memory, i.e. d = 0" and under (b) the alternative of "the process has a long memory, i.e. 0 < d < 1/2". Before we state our theorem, we set its context. Let (X t ) be a linear process of the form

X t = ∞ j=0 a j t-j (4)
where ( j ) are i.i.d. random variables with zero mean, variance σ 2 , and finite fourth moment η, and where ∞ j=0 a 2 j < ∞.

Theorem 1 Let X t be as defined in ( 4), Q n,m (s) be as defined in [START_REF] Brockwell | Time Series: Theory and Methods[END_REF] where

s ≥ 1 is fixed and m → ∞, m = o(n). (a) If X t is a short memory process in the sense that ∞ j=1 |a j | < ∞,
and that its spectral density f satisfies (1) with d = 0, then

Q n,m (s) d -→ Q(s),
where Q(s) has Gamma distribution with parameters (s, 1). (b) If X t is a long memory process in the sense that f is of the form (1) with 0 < d < 1/2, then

Q n,m (s) P -→ ∞.
Comments: (a) According to the convergence result in (a), the approximated critical region of the testing procedure

d = 0 versus 0 < d < 1/2 at significance level α is of the form {Q n,m (s) > Γ α (s)}
where Γ α (s) is the quantile of order 1 -α of Gamma distribution with parameters (s,1). This makes the test easy to implement as the critical values are available in all statistical softwares. (b) Under the condition b) of Theorem 1, our testing procedure is consistent, i.e. the power converges to 1 under any long memory alternative 0 < d < 1/2.

Proof of Part a) of Theorem 1. From (3) we get

Q n,m (s) = s j=1 I n (λ j )/f (λ j ) 1 m m i=1 I n,i (λ j )/f (λ j )
.

Applying Giraitis et al. Th 5.3.1 (2012), we have

I n (λ 1 ) f (λ 1 ) , • • • , I n (λ s ) f (λ s ) d → (E 1 , • • • , E s )
where E 1 , • • • , E s are i.i.d. exponentially distributed with mean 1, and since clearly Q(s)

d = E 1 +• • •+E s ,
using Slutsky Lemma, it will then be enough to show that, for each fixed j, as n → ∞,

1 m m i=1 I n,i (λ j ) f (λ j ) P -→ 1,
or equivalently (since f is continuous, and

λ j = 2πj n , then f (λ j ) → f (0)), 1 m m i=1 I n,i (λ j ) P -→ f (0) = 1 2π γ(0) + 2 ∞ k=1 γ(k) . (5)
Denoting

γn,i (h) = 1 i -|h| k=1+(i-1) X k X k+h ,
we can write

I n,i (λ) = 1 2π i k=1+(i-1) X k e iλk 2 = 1 2π i s,t=1+(i-1) X s X t e i(s-t)λ = 1 2π |h|< e -ihλ γn,i (h) = 1 2π γn,i (0) + 1 π -1 h=1 cos(λh)γ n,i (h).
Hence the LHS of ( 5) is made of two parts. Firstly,

1 2πm m i=1 γn,i (0) = 1 2π 1 m 1 m i=1 i k=1+(i-1) X 2 k = 1 2πn n u=1 X 2 u a.s. → γ(0) 2π , as n → ∞ (6)
by virtue of the ergodic theorem. Secondly, we prove the following L 2 convergence, as n → ∞

1 2πm m i=1 -1 h=1 γn,i (h) cos(λ j h) L 2 → 1 π ∞ k=1 γ(k). (7)
Observing that (γ n,1 (h), . . . , γn,m (h) are identically distributed),

1 π E 1 m m i=1 -1 h=1 γn,i (h) cos(λ j h) = 1 π -1 h=1 E(γ n,1 (h)) cos(λ j h) = 1 π -1 h=1 γ(h)(1 -h/ ) cos(λ j h) n→∞ -→ 1 π ∞ k=1 γ(k),
by the dominated convergence theorem, it suffices to prove that, as n → ∞,

Var 1 m m i=1 -1 h=1 γn,i (h) cos(λ j h) → 0. ( 8
)
The LHS of ( 8) can be written as

1 m 2 m s=1 m t=1 Cov -1 h=1 γn,s (h) cos(λ j h), - 1
h=1 γn,t (h) cos(λ j h) = 1 m m u=-m 1 - |u| m Cov -1 h=1 γn,1 (h) cos(λ j h) , -1 h=1 γn,1+|u| (h) cos(λ j h) = 1 m m u=-m 1 - |u| m -1 p=1 -1 q=1 cos(pλ j ) cos(qλ j )Cov γn,1 (p) , γn,1+|u| (q)
The first equality is due to the stationarity of the process

-1 h=1 γn,s (h) cos(λ j h) s .
Therefore it will be enough to show that for u ≥ 2,

p=1 q=1
Cov γn,1 (p), γn,1+u (q) → 0, as n, (or equivalently ) → ∞,

which is done in Lemma 1 in the appendix.

Proof of Part b) of Theorem 1. As Q n,m (s) is the sum of nonnegative terms, it will be enough to show that for each fixed j = 1, . . . , s

I n (λ j )/f (λ j ) 1 m m i=1 I n,i (λ j )/f (λ j ) P → ∞. (10)
Using [START_REF] Deo | Asymptotic theory for certain regression models with long memory errors[END_REF], the numerator in [START_REF] Lee | On the power of the KPSS test of stationarity against fractionallyintegrated alternatives[END_REF] converges in distribution to a non degenerated positive random variable. Therefore it will be enough to show that

1 m m i=1 I n,i (λ j ) f (λ j ) P → 0 as n → ∞. (11)
It is well known that if Z n ≥ 0 and E(Z n ) → 0 as n → 0 then Z n → 0 in probability. Therefore, since the variables involved in [START_REF] Lobato | A nonparametric test for I(0). The review of economic studies[END_REF] are nonnegative, it suffices to show that

1 m E m i=1 I n,i (λ j ) f (λ j ) → 0 as n → ∞. ( 12
)
Since I n,1 , . . . , I n,m are identically distributed, the LHS of ( 12) is simply equal to

E I n,1 (λ j ) f (λ j ) = -1 s,t=1 γ(s -t)e i(s-t)λ j f (λ j ) = h=- 1 -|h| γ(h) cos(λ j h) f (λ j ) . (13)
Under (1), f (λ) ∼ cλ -2d near zero (0 < d < 1/2), thus f (λ j ) ∼ cn 2d , and the numerator above is bounded by

h=- |γ(h)| ≤ C 1 h=1 h 2d-1 ∼ C 2 2d
for some positive constants c, C 1 , C 2 . Hence the ratio ( 13) is O ( /n) 2d → 0 as n → ∞ since d > 0 and n = 1 m → 0.

Simulation Studies

In this section we illustrate the performance of our test based on Q n,m (s) statistic and we compare it with the well known spectral domain test proposed by Lobato and Robinson (1998) and the time domain one based on VS statistic, briefly introduced below.

Lobato Robinson test (LR test).

In a one-dimensional case, Lobato and Robinson test statistic is LM = t 2 , where

t = √ m m j=1 ν j (I(λ j )) m j=1 (I(λ j ))
, and

ν j = log j - 1 m m j=1 log j.
Under certain conditions on the bandwidth m, if the spectral density is twice boundedly differentiable near 0 and d = 0, the statistic t converges in distribution to the standard normal distribution. Our approach is different since we use a blocking method rather than estimating f (0) and we do not assume such smoothness conditions on the spectral density near zero, except the continuity of g in ( 1), or on the tuning parameters except that the epoch length, (n) = o(n). In the simulation (Figure 3) we select for m the optimal value given in Lobato and Robinson (1998).

V/S test.

This test is defined as the ratio M n = V n /s 2 n,q , where

V n = 1 n 2 n k=1 (S * k ) 2 - 1 n n k=1 S * k 2 ,
with S * k = k j=1 (X j -Xn ), ŝ2 n,q = γ0 + 2 q j=1 1 -j q γj , and where γ is the sample covariance

γj = n -1 n-j i=1 (X i -Xn )(X i+j -Xn ), 0 ≤ j < n
where Xn is the sample mean of X 1 , . . . , X n . The null hypothesis of V/S test consists in the short memory case (which is our case (a) in theorem 1). Under some technical conditions on the bandwidth q and the fourth order cumulants (see [START_REF] Giraitis | Rescaled variance and related tests for long memory in volatility and levels[END_REF] for the exact hypotheses), M n converges to a non degenerated distribution with cdf

F V S (x) = F K (π √ x)
where F K is the asymptotic cdf of the standard Kolmogorov statistic. The authors suggest the use of q = n 1/3 for optimal results. This is the chosen value in our simulation (see Figure 3).

Choice of window parameters m and s in

Q n,m (s) defined in (3)
The choice of m is done to ensure that the empirical size will be close to the nominal size of the test. To do this we simulate independent copies of AR(1) and we evaluate for each sample the pvalue associated to Q n,m -test. Then we draw the empirical cumulative distribution function of pvalues. Under the null hypothesis this cdf should match the cdf of the uniform distribution.

Figure 1 represents these empirical cdfs of Q n,m (s) for several Gaussian AR(1) models. We take m = n b , b = .3, .4, .5, , .6 and n = 1000, 5000 and 10 000. Figure 1 shows a good fitting between the line y = x and the empirical cdf associated with the choice m = √ n. We also note that this choice remains good showing some robustness against deviation from normality of the innovations t (see Figure 3).

As for the parameter s, our simulation (with m = √ n) studies have shown that the choice of s depends on the sample size (See Figure 2). For small to moderate sample size, a small choice of s (such as s = 2, 3) would be preferable. This might be due to the fact that if e.x. we take n = 1000, each sub-periodogram is made of = 32 observations, and therefore only the very first Fourier frequencies (closest to frequency zero) will have a significant contribution in determining whether a data set come from a short or long memory process. We also note that when the sample size is quite large, we can still obtain a good empirical size while increasing s. Figure 3 again shows e.x., that a good choice of s would be s ≤ 5 for n = 5000 and s ≤ 10 for n = 10, 000 We suggest that if the sample is very large, then we should slihhtly increase s.

Non Gaussian innovations

Simulations have shown that under normality assumption LR test tends to outperform all time domain tests such as V/S, KPSS, RS etc. Here we compare our Q n,m (s) test with LR test and V/S test under the null hypothesis when the innovations of AR(1) are non Gaussian. (we generated innovations from heavily skewed exponential and log normal distributions, but we only reported the results for exponential distribution. Figure 3 shows a lack of robustness to non-normality of V/S and LR tests in the sense that these tests seem to suffer from high empirical size (the ratio of rejecting the null is much bigger than the nominal level α = .05, .1 etc). On the opposite, Q n,m (s) test seems to have a more reasonable empirical size (very much close to the nominal level α = .05, .1 etc). Similar conclusions hold for innovations with log-normal distributions as well.

Power of Q n,m (s) test.

Figure 4 illustrates the behavior of the empirical power of this test when s varies. We only represent the graphs for a non-Gaussian case (innovations with log normal distribution) but there are no noticeable differences with the Gaussian case. As expected, it shows an increase in the power function with the parameter s. Since LR and V/S empirical sizes are quite high, as seen in the previous section, as compared to Q n,m (s) test, the former tests will tend to be more powerful than Q n,m (s) test especially in Gaussian case. This is very much anticipated and for this reason we did not report the power comparison graphs between V/S and LR on one side and Q n,m (s) on the other side.

In conclusion, our findings confirm that, mainly due to its relatively easy implementation and large sample properties, frequency-domain approach should be used more often in combination with classical time domain methods when it comes to testing for long memory or for unit root (as AR(1) coefficient tends to 1) in the time series analysis. γ(p)γ(q) + γ(s -t + u)γ(s -t + u + q -p) +γ(s -t + u + q)γ(s -t + u -p)

+(η -3)σ 4 ∞ i=0
a i a i+p a i+s-t+ u a i+s-t+ u+q -γ(p)γ(q) = 1

-p+1

h=1+q-

1 -|h| T ,u (h, p, q), where T ,u (h, p, q) = T

,u (h, p, q) + T

,u (h, p, q) + T

,u (h, p, q), with T

,u (h, p, q) = γ( u -h)γ( u -h + q -p), T

,u (h, p, q) = γ( u -h + q)γ( u -h -p), and

T (3)
,u (h, p, q) = (η -3)σ 4 ∞ i=0 a i a i+p a i+h+ u a i+h+ u+q .

Next for u ≥ 2, putting k = q -p,

,u (h, p, q) → 0, as → ∞.

As for the last term, permuting the order of summation, and using the variable change s = i + h + u, and without loss of generality, considering the coefficients a i ≥ 0.

h=-T

,u (h, p, q) = (η -3)σ 4

,u (h, p, q) = o( ) → 0, as → ∞, which completes the proof of the lemma.

∞

  h=1 cos(λh)γ(h) , and reciprocally γ(h) = π -π e ihλ f (λ)dλ = π -π cos(λh)f (λ)dλ.

Figure 1 :

 1 Figure 1: P-values of Q n,m (2); 5000 independent replications of AR(1) model with AR coefficient a = -0.5, 0.5, 0.8 and Gaussian innovations.

Figure 2 :

 2 Figure 2: P-values of Q n, √ n (s); 5000 independent replications of AR(1) model with AR coefficient a = 0.5 and Gaussian innovations.

Figure 3 :

 3 Figure 3: P-values of Q n, √ n (2); 5000 independent replications of AR(1) model with AR coefficient a = -0.5, 0.5, 0.8 and non gaussian innovations: t = ξ t -1 where ξ t are i.i.d. standard exponentials

  h, p, q) = h=-k=-1 -|k| |γ( u -h)γ( u -h + k)| ≤ h=-k=-|γ( u -h)γ( u -h + k)|since the covariances are summable in short memory. Similar computation shows also that 1 p=1 q=1 h=-T

  s+q = γ(p)o(1), as → ∞, and therefore, as the covariances are summable in short memory, 1 p=1 q=1 h=-T

Appendix

In this appendix we state and prove the convergence (9) used in the proof of Theorem 1-a.

Lemma 1 Under the hypotheses of Theorem 1-a,

Cov γn,1 (p), γn,1+u (q) → 0, as n, (or equivalently ) → ∞.

Proof We recall that σ 2 is the variance of t and η is its fourth moment. We have with the variable change s = z -u, Cov γn,1 (p), γn,1+u (q) = 1

[E[X t X t+p X t+s-t+ u-p+p X t+s-t+ u-p+p+q ] -γ(p)γ(q)] .


Amelia Carolina Sparavigna

Image segmentation applied to micrographs of microcellular plastics

Keywords: Microcell sizing, image segmentation, image processing, microcellular plastics

The paper is proposing the use of a method of image segmentation to the study of microcellular plastics. The segmentation is based on a thresholding which creates a binary (black and white) image of the micrograph. The binary image is divided in super-pixels which correspond to the microcells of the material. From the areas of the super-pixels it is easy to evaluate the size of the cells.

Here we are proposing the use of the image segmentation, one of the fundamental methods of the image processing, to the study of microcellular plastics. The microcellular plastic, also known as microcellular foam, is a form of material fabricated to contain tiny bubbles less than 50 microns in size. The bubbles are formed by dissolving a gas under high pressure into a polymer, to obtain an almost uniform arrangement of the gas bubbles [START_REF] Suh | Impact of microcellular plastics on industrial practice and academic research[END_REF]. The plastic that is obtained has good mechanical properties, with a reduced use of material; in it, the density of the foam is ranging between 5% to 99% of the pre-processed plastic [2].

Recently, researches made at the Indian Institute of Technology in Delhi have been published on new technologies based on ultrasounds for the development of high quality microcellular foams [START_REF] Gandhi | Ultrasound assisted cyclic solid-state foaming for fabricating ultra-low density porous acrylonitrile-butadiene-styrene foams[END_REF][START_REF] Gandhi Abhishek | Ultrasound-Induced Nucleation in Microcellular Polymers[END_REF]. In [START_REF] Vv | Wikipedia[END_REF], we can see a micrograph of a sample obtained by this Institute. It could be interesting to evaluate the distribution, in size, of the cells we observe in the micrograph. To this purpose, an image segmentation can be used. Actually, by means of an image segmentation we can obtain a segmented image made by super-pixels, that, in the case we are considering, correspond to the microcells of the plastic material. The evaluation of the area (in pixels) of the superpixels give us the possibility of estimating the size of the cells.

In image processing, a segmentation is a process of partitioning an image into multiple sets of pixels, defined as super-pixels, in order to have a representation which is simpler than the original one and more useful to the following desired analyses [START_REF] Shapiro | Computer vision[END_REF][START_REF] Pham | Current methods in medical image segmentation[END_REF][START_REF] Forghani | Parameter optimization of improved fuzzy c-means clustering algorithm for brain MR image segmentation[END_REF][START_REF] Vv | Wikipedia[END_REF]. The typical use of the image segmentation is that of locate objects, or domains, and boundaries among them. Specifically, the segmentation is a process of assigning a label to every pixel in an image, such that the pixels having the same label share certain characteristics [START_REF] Vv | Wikipedia[END_REF]. As a consequence, the result of the segmentation is a set of "segments", also known as "super-pixels", that are covering the whole image.

Several methods exist for segmentation, as we can appreciate from [START_REF] Vv | Wikipedia[END_REF]. In [START_REF] Sparavigna | A method for the segmentation of images based on thresholding and applied to vesicular textures[END_REF], we have proposed a method based on thresholding, a method aimed to be used in the analysis of the vesicular textures, the textures we can encounter in geology, materials science and physics. The vesicular textures in images or micrographs are features that evidence the presence of cavities or holes in the samples; these textures look like those displayed by the volcanic rocks when are pitted with many cavities (known as vesicles) at their surfaces and inside. In such cases, the vesicles are made by gas escaping from cooling lava.

Another example of the use of the image segmentation has been discussed in [START_REF] Sparavigna | Analysis of a natural honeycomb by means of an image segmentation[END_REF], for the case of a natural honeycomb structure. In [START_REF] Sparavigna | Analysis of a natural honeycomb by means of an image segmentation[END_REF] we plotted the distribution of the super-pixels, and consequently, the frequencies of the areas of the cells that we see in the image of the honeycomb. In the Figure 1, the three steps of segmentation by thresholding are illustrated [START_REF] Sparavigna | A method for the segmentation of images based on thresholding and applied to vesicular textures[END_REF][START_REF] Sparavigna | Analysis of a natural honeycomb by means of an image segmentation[END_REF]. We start from the original image, converted in grey tones, as in the panel A of the Figure . Then we use GIMP (GNU Image Manipulation Program for X Windows systems) for a simple thresholding to have the black and white image, reproduced in panel B (in fact, a thresholding based on the entropy of image is also possible [START_REF] Sparavigna | Tsallis entropy in bi-level and multi-level image thresholding[END_REF][START_REF] Sparavigna | Shannon, Tsallis and Kaniadakis entropies in bi-level image thresholding[END_REF][START_REF] Sparavigna | The Tsallis Entropy and the Segmentation of Images[END_REF]). On the binary black and white image, we apply a Fortran program for segmentation [START_REF] Sparavigna | A method for the segmentation of images based on thresholding and applied to vesicular textures[END_REF]. In the panel C, we show the super-pixels having different colour tones. Since each black pixel of the original grey-tone image has the label of the super-pixel to which it belongs, we can easily do some calculations. For instance, we can calculate the number of the super-pixels. Then, we can give the number of pixels contained in each super-pixel. In this manner we can measure the area of each cell of the honeycomb lattice, as given in the panel D of the Figure 1. Of course, we can do the same for the case of a microcellular plastic (Figure 2). The original image is the micrograph given in [START_REF] Vv | Wikipedia[END_REF]. As we can see from the example in the Figure 2, some small super-pixels can appear, due to the segmentation of the regions that we find between the large cells in the image. To reduce their role in the counting of the distribution of the super-pixels, the binary image we obtain after the thresholding of the grey-level image is further subjected to a processing. This processing is made by the GIMP generic filter "dilate", followed by the generic filter "erode" [START_REF]Details on GIMP filters "dilate" and "erode[END_REF]. The segmentation we obtain is given in the Figure 3. If we consider the segmentation and the super-pixels as given in the Figure 3, we can determine the distribution of the microcell cross-sections, by counting them according to their area (in pixels) within intervals spaced of 50 pixels. We obtain the distribution given in the Figure 4 (the super-pixels having an area less than 25 pixels are not considered). Figure 4 shows that a few polyhedra having a large size exist, but many polyhedra with a cross-section between 100 and 400 pixels are found in the micrograph. It means that we have a large number of cells with size ranging from 6 to 11 pixels (in the case that we have also a scale on the recorded micrograph, we can give the size in microns). In the Figure 5 and 6, we are proposing another example.

In this paper we have proposed the use of an image segmentation in the study of microcellular plastics. The method of segmentation is based on thresholding. We have also filtered the image for avoiding a spurious number of very small super-pixels. Further studies are necessary to link the distribution of the super-pixels, and then of the microcells, to the mechanical properties of the material.

Figure 1 :

 1 Figure 1: The grey-tone image (240 x 240 pixels) is given in the panel A (Courtesy: Audrius Meskauskas, Wikipedia). In B, the corresponding black and white image is given, as obtained after a thresholding. In C, the result of the segmentation is shown. Each super-pixel is given in different colour tone. In D, some crosssectional area (in pixels) of the honeycomb cells are given.

Figure 2 :

 2 Figure 2: The grey-tone image (240 x 240 pixels) is given in the left panel (Microcellular Plastic Micrograph developed at Indian Institute of Technology, Delhi. Courtesy: Gandhi.iitdelhi, Wikipedia). In the middle, the corresponding black and white image obtained after a thresholding. On the right, the result of the segmentation is given. Each super-pixel is rendered in different colour tone.

Figure 3 :

 3 Figure 3: The whole micrograph of the Figure 2 and its segmentation. (Microcellular Plastic Micrograph developed at Indian Institute of Technology, Delhi. Courtesy: Gandhi.iitdelhi, Wikipedia). The area of the image is 600 x 384 pixels.

Figure 4 :

 4 Figure 4: Distribution of the super-pixels.

Figure 5 :

 5 Figure 5: An image of microcellular plastic and its segmentation. (Microcellular Plastic Micrograph developed at Indian Institute of Technology, Delhi. Courtesy:Gandhi.iitdelhi, Wikipedia, https://commons.wikimedia.org/wiki/File:Cyclic_Microcellular_Foam.tif). The area of the image is 600 x 376 pixels.

Figure 6 :

 6 Figure 6: Distribution of the super-pixels.


M Peigné

A Sambusetti

Entropy Rigidity of negatively curved manifolds of finite volume

Keywords: AMS classification : 53C20, 37C35 Negative curvature, entropy, length spectrum, Bowen-Margulis measure

We prove the following entropy-rigidity result in finite volume: if X is a negatively curved manifold with curvature -b 2 ≤ K X ≤ -1, then Ent top (X) = n -1 if and only if X is hyperbolic. In particular, if X has the same length spectrum of a hyperbolic manifold X 0 , the it is isometric to X 0 (we also give a direct, entropy-free proof of this fact). We compare with the classical theorems holding in the compact case, pointing out the main difficulties to extend them to finite volume manifolds.

Introduction

The problem of length spectrum rigidity of Riemannian manifolds has a long history. The fact that, in negative curvature (even in constant curvature), the collection of the lengths of all closed geodesics, together with all multiplicites, does not determine the metric is well known since [START_REF] Vigneras | Varie tés Riemanniennes isopectrales et non isométriques[END_REF]. On the other hand, on a compact, negatively curved surface X, the metric is determined up to isometry by the marked length spectrum (that is, the map L : C( X) → R associating to each free homotopy class of loops in X the length of the shortest geodesic in the class); this was proved by Otal [START_REF] Otal | Le spectre marqué des longueurs des surfaces à courbure négative[END_REF] and, independently, by Croke [START_REF] Croke | Rigidity for surfaces of non-positive curvature[END_REF]. The same is true in dimension n ≥ 3 for any compact, locally symmetric manifold X0 of negative curvature: the locally symmetric metric on X0 is determined, among all negatively curved metrics, by its marked length spectrum. This is consequence of Besson-Courtois-Gallot's solution of the minimal entropy conjecture and of the fact, proved by Hamenstadt [START_REF] Hamenstädt | Cocycles, symplectic structures and intersection[END_REF], that if a compact, negatively curved manifold X has the same marked length spectrum as a compact, locally symmetric space X0 , then vol( X) = vol( X0 ) 1 .

1 By [START_REF] Hamenstädt | Time preserving conjugacies of geodesic flows[END_REF], two compact, negatively curved manifolds having the same marked length spectrum have C 0 -conjugated geodesic flow; moreover, if a compact manifold X has geodesic flow which is C 0conjugated to the flow of a manifold X0 whose unitary tangent bundle has a C 1 -Anosov splitting (e.g., a locally symmetric space), then X has the same volume as X0, see [START_REF] Hamenstädt | Cocycles, symplectic structures and intersection[END_REF]. The fact that, for compact manifolds, volume is preserved under C 1 -conjugacies is much easier and relies on Stokes' formula, cp. [START_REF] Croke | Conjugacy and rigidity for manifolds with a parallel vector field[END_REF].

Less seems known about the length rigidity of negatively curved, finite-volume manifolds: most generalizations are not straightforward, and seem to require additional assumptions (such as bounds on the curvature and on its derivatives, or the finiteness of the Bowen-Margulis measure); we will try to point out some of these difficulties throughout the paper. For instance, the fact that having the same marked length spectrum implies the existence of a C 0 -conjugacy of the geodesic flow, would certainly require some new arguments for finite-volume manifolds2 . Theorem 1.1 Let X be a finite volume n-manifold with pinched, negative curvature -b 2 ≤ K X ≤ -1 which is homotopically equivalent to a locally symmetric manifold X0 , with curvature normalized between -4 and -1. If X and X0 have same marked length spectrum, then they are isometric.

The proof of this is probably known to experts and follows a classical scheme: one can construct a Γ-equivariant map f : X → X 0 between the universal coverings, which induces a homeomorphism between the boundaries and preserves the cross-ratio; then, the conclusion stems, for instance, from Bourdon's result [START_REF] Bourdon | Sur le birapport au bord des CAT (-1)-espaces[END_REF] on Möbius embeddings from locally symmetric to CAT (-1)-spaces. However, the main difficulty, in the case of finite volume manifolds, is to show that f is a quasi-isometry, the quotients X and X0 being non-compact; we will give a short proof of this fact in §3, by way of example, to measure the difference from the compact case.

It is tempting to approach the above problem by using a finite-volume version of Besson-Courtois-Gallot's inequality, given by Storm [START_REF] Storm | The minimal entropy conjecture for nonuniform rank one lattices[END_REF]; however, notice that Storm's inequality Ent( X) n V ol( X) ≥ Ent( X0 ) n V ol( X0 ) concerns the volume entropy3 of X, and not the topological entropy Ent top ( X) of the geodesic flow on U X. Recall that for compact, negatively curved manifolds, one always has Ent(X) = Ent top ( X), but for finite-volume manifolds Ent( X) is generally strictly greater than Ent top ( X) (cp. [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF], [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF]); on the other hand, the topological entropy always equals the critical exponent of the group Γ = π 1 (X) acting on the universal covering:

δ Γ := lim R→∞ 1 R ln #{γ ∈ Γ | d(x, γx) ≤ R}
as proved in [START_REF] Otal | Principe variationnel et groupes kleiniens[END_REF]. Then, the volume entropy Ent( X) is not preserved, a priori, by the condition of having same marked length spectrum, or by a conjugacy of the geodesic flows. Moreover, it is not clear whether, for finite volume manifolds, the volume is preserved under a conjugacy of the flows 4 . The upper bound on the curvature K X ≤ -1 in Theorem 1.1 seems unreasonably strong, as it implies, when X0 is hyperbolic, that X has marked length spectrum which is asymptotically critical: that is, its exponential growth rate δ Γ is greater than or equal to the corresponding exponential growth rate for X0 (cp. [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF], Lemma 4.1).

We expect that the same result holds without curvature bounds, but, even in this weaker form, we were unable to find a proof of this result in literature.

The knowledge of the full marked length spectrum can be relaxed, as we show in the following result (which implies Theorem 1.1 in the real hyperbolic case): Theorem 1.2 Let X be a finite volume n-manifold with pinched, negative curvature -b 2 ≤ K X ≤ -1. Then Ent top ( X) ≥ n -1, and the equality Ent top ( X) = n -1 holds if and only if X is hyperbolic.

The entropy characterization of constant curvature (and locally symmetric) metrics has been declined in many different ways so far: in the compact case, the above theorem is due to Knieper (see [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF], where this result is not explicitly stated, but can be established following the argument of the proof of Theorem 5.2.); see also [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF], and [START_REF] Bonk | Rigidity for quasi-Möbius group actions[END_REF] for a proof in the convex-cocompact case.

We want to stress here that a basic difference between Theorem 1.2 (or, more precisely, their compact versions in [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF], [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF], [START_REF] Bonk | Rigidity for quasi-Möbius group actions[END_REF]) and the celebrated entropy characterization of Hamenstädt [START_REF] Hamenstädt | Entropy-rigidity of locally symmetric spaces of negative curvature[END_REF] of locally symmetric metrics, with same curvature normalization, is the lack of any locally symmetric manifold X0 of reference homotopically equivalent to X. Actually, the characterization given by Theorem 1.2 is very particular to constant curvature spaces and it does not generalize, as it is, to locally symmetric spaces: indeed, it is easy to construct compact, pinched, negatively curved manifolds with -b 2 ≤ K X ≤ -1 having same entropy as, let's say, the complex hyperbolic space, but which are not complex hyperbolic.

The same difference holds with the existing, finite volume versions of Besson-Courtois-Gallot's characterization of locally symmetric spaces, in particular with Boland-Connell-Souto's papers [START_REF] Boland | Volume rigidity for finite volume manifolds[END_REF] and Storm's [START_REF] Storm | The minimal entropy conjecture for nonuniform rank one lattices[END_REF]: these two works, together, imply that if a finite volume manifold X with curvature K X ≤ -1 has volume entropy Ent( X) = n -1, then it is hyperbolic, provided that one knows beforehand that X is homotopically equivalent to a hyperbolic manifold X0 . Besides the difference between volume and topological entropy stressed above, this strong supplementary topological assumption on X is not made in Theorem 1.2.

Let us also point out that Knieper's approach in [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF] does not allow to deduce the above characterization in the finite volume case. Although G. Knieper's horospherical measure µ H can be perfectly defined in this context (following §3 of [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF]), it can easily be infinite, as well as the Bowen-Margulis measure µ BM : given a finite volume surface X with convergent fundamental group Γ and with a cusp whose metric, in horospherical coordinates, writes as A 2 (t)dx 2 + dt 2 , it is not difficult to show that µ H is infinite as soon as [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF]). Therefore, all formulas in [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF] relating Ent top ( X) to the trace of the second fundamental form of unstable horospheres need to be justified in some other way 5 .

∞ 0 e δ Γ t A(t)dt = +∞ (cp. examples in §3,
On the other hand, we will give in §4 a proof of Theorem 1.2 using the barycenter method, initiated by Besson-Courtois-Gallot in [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]- [START_REF] Besson | Lemme de Schwarz réel et applications géométriques[END_REF], together with some careful estimates of the Patterson-Sullivan measure, which will not need neither the finiteness of µ BM (or µ H ) nor the conservativity of the geodesic flow with respect to µ BM .

Also, notice that if we drop the assumption K X ≥ -b 2 in Theorem 1.2, the manifold X might as well be of infinite type (i.e. with infinitely generated fundamental group, or even without any cusp, see examples in [START_REF] Nguyen Phan | On finite volume, negatively curved manifolds[END_REF]), hence very far from being a hyperbolic manifold of finite-volume.

Notations.

Given functions f, g : R + → R + , we will systematically write f

C ≺ g (or g C ≻ f ) if there exists C > 0 and R 0 > 0 such that f (R) ≤ Cg(R) for R > R 0 . We write f C ≍ g when g C ≺ f C ≺ g for R ≫ 0 (
or simply f ≍ g and f ≺ g when the constants C and R 0 are unessential) 2 Geometry at infinity in negative curvature Throughout all the paper, X will be a n-dimensional, complete, simply connected manifold with strictly negative curvature -b 2 ≤ K X ≤ -1.

Let X(∞) the ideal boundary of X: for x, y ∈ X and ξ ∈ X(∞), we will denote by with center ξ and passing through x. We will denote by d ξ the horospherical distance between two points on a same horosphere centered at ξ, and we define the radial semiflow (ψ ξ,t ) t≥0 in the direction of ξ as follows: for any x ∈ X, the point ψ ξ,t (x) lies on the geodesic ray [x, ξ[ at distance t from x.

Finally, recall that for any fixed x ∈ X, the Gromov product between two points ξ, η ∈ X(∞), ξ = η, is defined as

(ξ|η) x = b ξ (x, y) + b η (x, y) 2
where y is any point on the geodesic ]ξ, η[ joining ξ to η; as )x defines a distance on X(∞), which we will call the visual distance from x, cp. [START_REF] Bourdon | Structure conforme au bord et flot géodésique d'un CAT(-1)-espace[END_REF]. Accordingly, the cross-ratio on X(∞) 4 is defined as

K X ≤ -1, the expression D x (ξ, η) = e -(ξ|η
[ξ 1 , ξ 2 , ξ 3 , ξ 4 ] = D x (ξ 1 , ξ 3 )D x (ξ 2 , ξ 4 ) D x (ξ 1 , ξ 4 )D x (ξ 2 , ξ 3 ) = lim p 1 ,p 2 ,p 3 ,p 4 ∈X (p 1 ,p 2 ,p 3 ,p 4 )→(ξ 1 ,ξ 2 ,ξ 3 ,ξ 4 ) e d(p 1 ,p 3 )+d(p 2 ,p 4 )-d(p 1 ,p 4 )-d(p 2 ,p 3 )
for all ξ 1 , ξ 2 , ξ 3 , ξ 4 ∈ X(∞), and it is easily seen that it is independent from the choice of the base point x, cp. [START_REF] Otal | Sur la géométrie symlectique de l'èspace des géodésiques d'une variété à courbure négative[END_REF], [START_REF] Bourdon | Structure conforme au bord et flot géodésique d'un CAT(-1)-espace[END_REF].

We will repeatedly make use of the following, classical result in strictly negative curvature: there exists ǫ(ϑ) = log( 21-cos ϑ ) such that any geodesic triangle xyz in X making angle ϑ = ∠ z (x, y) at z satisfies:

d(x, y) ≥ d(x, z) + d(z, x) -ǫ(ϑ).
(1)

On the geometry of finite volume manifolds

Consider a lattice Γ of X. The quotient manifold X = Γ\X has finite volume, it is thus a geometrically finite manifold which admits some particular decomposition which we now recall. The following classical results are due to B. Bowditch [START_REF] Bowditch | Geometrical finiteness with variable negative curvature[END_REF], and we state them in the particular case of finite volume manifolds :

(a) the limit set of L(Γ) of Γ is the full boundary at infinity X(∞) and is the disjoint union of the radial limit set L rad (Γ) with finitely many orbits of bounded parabolic fixed points L bp (Γ) = Γξ 1 ∪ . . . ∪ Γξ l ; this means that each ξ i ∈ L bp (Γ) is the fixed point of some maximal parabolic subgroup P i of Γ, acting co-compactly on X(∞) \ {ξ i };

(b) (Margulis' lemma) there exist closed horoballs H ξ 1 , . . . , H ξ l centered respectively at ξ 1 , . . . , ξ l , such that γH ξ i ∩ H ξ j = ∅ for all 1 ≤ i, j ≤ l and all γ ∈ Γ \ P i ;

(c) The finite volume manifold X can be decomposed into a disjoint union of a compact set K and finitely many "cusps" C1 , ..., Cl : each Ci is isometric to the quotient of H ξ i by a corresponding maximal bounded parabolic group P i . We refer to K and to C = ∪ i Ci as to the thick part and the cuspidal part of X.

For any fixed x ∈ X, let D = D(Γ, x) the Dirichlet domain of Γ centered at x; this is a convex fundamental subset of X, and we may assume that D contains the geodesic rays [x, ξ i [. Each parabolic group P i acts co-compactly on the horosphere ∂H ξ i which bounds the horoball H ξ i ; setting

S i = D ∩ ∂H ξ i and C i = D ∩ H ξ i ≃ S i × R + , the fundamental domain D can be decomposed into a disjoint union D = K ∪ C 1 ∪ • • • ∪ C l ( 2
)
where K is a convex, relatively compact set containing x in its interior and projecting to the thick part K of X , while C i and S i are, respectively, connected fundamental domains for the action of P i on H ξ i and ∂H ξ i , projecting respectively to Ci and Si .

Growth of parabolic subgroups

The subgroups P 1 , s, P l will play a crucial role in the sequel; the growth of their orbital functions is best expressed by introducing the horospherical area function.

Let us recall the necessary definitions:

Definition 2.

(Horospherical Area)

Let P be a bounded parabolic group of isometries of X fixing ξ ∈ X(∞): that is, P acts cocompactly on X(∞) {ξ} (as well as on every horosphere centered at ξ). Given x ∈ X, let S x be a fundamental, relatively compact domain for the action of P on ∂H ξ (x): the horospherical area function of P is the function

A P (x, R) = vol [P \ψ ξ,R (∂H ξ (x))] = vol [ψ ξ,R (S x )]
where vol denotes the Riemannian measure of horospheres.

Remark 2.2 When -b 2 ≤ K X ≤ -a 2 < 0, well-known estimates of the differential of the radial flow (cp. [START_REF] Heintze | Geometry of horospheres[END_REF]) yield, for any

t ∈ R and v ∈ T 1 X e -bt v ≤ dψ ξ,t (v) ≤ e -at v (3)
Therefore we deduce that, for any ∆ > 0, e -(n-1)b∆ ≤ A P (x, R + ∆)

A P (x, R) ≤ e -(n-1)a∆ (4)
The following Proposition shows how the horospherical area A P is related to the orbital function of P , cp. [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF]:

Proposition 2.3 Let P be a bounded parabolic group of X fixing ξ, with diam(S x ) ≤ d.
There exist R 0 and ∆ 0 only depending on n, a, b, d and constants C = C(n, a, b, d) and a,b,d,∆) such that, for any R ≥ b ξ (x, y) + R 0 and any ∆ > ∆ 0 , the numbers v P (x, y, R) and v ∆ P (x, y, R) of orbit points of P y falling, respectively, in the balls B(x, R) and in the annuli A ∆ (x, R) satisfy:

C ′ = C ′ (n,
v P (x, y, R) = {p ∈ P | d(x, py) < R} C ≍ A -1 P x, R + b ξ (x, y) 2 v ∆ P (x, y, R) = {p ∈ P | R - ∆ 2 ≤ d(x, py) ≤ R + ∆ 2 } C ′ ≍ A -1 P x, R + b ξ (x, y) 2 .

Length spectrum and rigidity

This section is devoted to the proof of Theorem 1.1.

Let Γ be the fundamental group of the manifolds X and X0 , acting by isometries on their Riemannian universal coverings X and X 0 respectively. We will construct a Γ-equivariant homeomorphism f ∞ : ∂X(∞) → X 0 (∞) and apply the following:

Theorem 3.1 (cp. [START_REF] Bourdon | Sur le birapport au bord des CAT (-1)-espaces[END_REF]) Let X be a CAT(-1)-space and X 0 a symmetric space of rank one, with curvature -4 ≤ K X 0 ≤ 1. Assume that f ∞ : X(∞) → X 0 (∞) is a Γ-equivariant homeomorphism which preserves the cross-ratio: then there exists a Γ-equivariant isometry f : X → X 0 whose extension on X(∞) coincides with f ∞ .

For this, we fix x ∈ X and x 0 ∈ X 0 and consider the natural Γ-equivariant bijection φ : Γx → Γx 0 . The main difficulty here is to show the following:

Proposition 3.2
The map φ is a quasi-isometry between the orbits, with respect to the distances induced by the Riemannian distances of X and X 0 respectively.

We assume Proposition 3.2 for a moment. Since X and X0 have finite volume, the limit set of Γ coincides with the full boundaries X(∞) and X 0 (∞) = S n-1 and the map φ extends to a bi-Hölder and Γ-equivariant homeomorphism f ∞ between these boundaries, endowed with their natural visual metric from x and x 0 . Now, the fact that X and X0 have the same marked length spectrum implies that f ∞ preserves the cross ratio; this follows for instance from [START_REF] Otal | Sur la géométrie symlectique de l'èspace des géodésiques d'une variété à courbure négative[END_REF]. For the sake of completeness, we will give a proof of this fact at the end of this section (Proposition 3.5), based on an argument from [START_REF] Kim | Marked length rigidity of rank one symmetric spaces and their product[END_REF] (where the same is proved for symmetric spaces). We conclude by 3.1 that there exists an isometry between the quotients X and X0 . 2

Proof of Proposition 3.2. Let us first show that there exists λ > 1 such that, for all γ ∈ Γ, we have

d 0 (x 0 , γx 0 ) ≤ λd(x, γx) + λ. (5)
Consider the decomposition of X described in subsection 2.1: we denote by H the set of pairwise disjoint horoballs which project on the cuspidal part of X, so that K := X \ ∪ H∈H H = ΓK is the subset of X projecting to the thick part K of X. We assume that d(H, H ′ ) ≥ 1 for any H = H ′ in H, and set diam(K) = D.

For any γ ∈ Γ, the geodesic segment [x, γx] intersects at most N ≤ d(x, γx) distinct horoballs H ∈ H and can be decomposed as

[x, γx] = [x + 0 , x - 1 ] ∪ [x - 1 , x + 1 ] ∪ • • • ∪ [x + N -1 , γx - N ] where x + 0 = x, x - N = x, and where [x - i , x + i ] is equal to [x, γx] ∩ H i for some horoball H i ∈ H and each [x + i , x - i+1 ] is included in K.
Then, there exist elements g i ∈ Γ and

p i ∈ P 1 ∪ • • • ∪ P l , for 1 ≤ i ≤ N -1, with g N := γ, such that x - i ∈ g i K, x + i ∈ g i p i K; moreover, set γ i := p -1 i-1 g -1 i-1 g i for 1 ≤ i ≤ N with the convention p 0 = g 0 = 1. Notice that all the geodesics [x, γ i x] are included in a D ′ = D ′ (D)-neighbourhood of K: in fact, the length of the broken geodesic [x + i , g i p i x]∪[g i p i x, g i+1 x]∪[g i+1 x, x - i+1 ] exceeds the length of [x + i , x - i+1
] at most of 2D, so (the curvature being bounded above by -1) it stays D ′ (D) close to [x + i , x - i+1 ]; by construction this last segment does not enter any horoball of H, so [g i p i x, g i+1 x] and [x,

γ i+1 x] = p -1 i g -1 i [g i p i x, g i+1 x] stay in the D ′ -neighbourhood of K. Now we have γ = γ 1 p 1 γ 2 • • • γ N -1 p N -1 γ N , so d 0 (x 0 , γx 0 ) ≤ N i=1 d 0 (x 0 , γ i x 0 ) + N -1 i=1 d 0 (x 0 , p i x 0 ). (6)
On the other hand

d(x, γx) = N i=1 d(x + i-1 , x - i ) + N -1 i=1 d(x - i , x + i ) ≥ N -1 i=1 d(g i-1 p i-1 x, g i x) + N -1 i=1 d(g i x, g i p i x) -4(N -1)D = N i=1 d(x, γ i x) + N i=1 d(x, p i x) -4(N -1)D which in turn yields, as N ≤ d(x, γx), N i=1 d(x, γ i x) + N i=1 d(x, p i x) ≤ (1 + 4D)d(x, γx). (7)
To obtain inequality [START_REF] Bourdon | Structure conforme au bord et flot géodésique d'un CAT(-1)-espace[END_REF], it is thus sufficient to check that it holds for each γ i and p i which appears in the sums ( 6) and [START_REF] Bowditch | Geometrical finiteness with variable negative curvature[END_REF]. This is proved in the two following lemmas:

Lemma 3.3 For any D ′ > 0, there exists C > 0 such that

d 0 (x 0 , γx 0 ) ≤ C d(x, γx) + C
for any γ ∈ Γ such that [x, γx] lies in the D ′ -neighbourhood of K.

Lemma 3.4 There exists C ′ > 0 such that, for any parabolic isometry

p ∈ P 1 ∪ • • • ∪ P l d(x, px) C ′ ≍ d 0 (x 0 , px 0 ). ( 8
)
Switching the roles of (X, d) and (X 0 , d 0 ), we obtain the opposite inequality d 0 (x 0 , γx 0 ) ≤ λd(x, γx) + λ, which concludes the proof of Proposition 3.2.

2

Proof of Lemma 3.3. Let γ ∈ Γ such that [x, γx] lies in the D ′ -neighbourhood of K, and let

x 0 = x, x N = γx and x 1 , • • • , x N -1 be the points on the geodesic segment [x, γx] such that d(x, x i ) = iD for 0 ≤ i ≤ N -1, with N -1 = [d(x, γx)]. There exist isometries h 0 = 1, h 1 , • • • , h N -1 , h N = γ in Γ such that d(x i , h i x) ≤ D + D ′ ; setting k i = h -1 i-1 h i , we then have γ = k 1 k 2 • • • k N . Now, for any 1 ≤ i ≤ N , we have d(x, k i x) ≤ 1 + D + D ′ ; so every k i belongs to the finite set B := {k ∈ Γ | d(x, kx) ≤ 1 + D + D ′ }. Setting C := max{d 0 (x 0 , kx 0 ) | k ∈ B}, we obtain d 0 (x 0 , γx 0 ) ≤ N i=1 d 0 (x 0 , k i x 0 ) ≤ N C ≤ Cd(x, γx) + C. 2 Proof of Lemma 3.4.
Let us first notice that if p ∈ Γ acts on X as a parabolic (resp. a hyperbolic) isometry, then it acts in the same way on X 0 : actually, the infimum of the length of curves in X in the free homotopy class of a parabolic element p is 0 and this condition is preserved since X and X0 have the same length spectrum. Then, let ξ 1 , ..., ξ l ∈ X(∞) be the fixed points of the maximal parabolic subgroups P 1 , .., P l of Γ such that the geodesic rays [x, ξ i [ are included in the Dirichlet domain D, as described in the subsection §2.1, and call ξ ′ i the corresponding parabolic fixed points of X 0 (∞); in order to simplify the notations, we set P = P i , ξ = ξ i and ξ ′ = ξ ′ i . Fix a finite generating set S for P and let | • | S be the corresponding word metric. As P acts cocompactly by isometries on (∂H ξ (x), d ξ ) and on (∂H ξ ′ (x 0 ), d ξ ′ ) we know that these metric spaces are both quasi-isometric to (P, |•| S ). In particular, there exists a constant c > 0 such that, for any p ∈ P

1 c d ξ (x, px) -c ≤ d ξ ′ (x 0 , px 0 ) ≤ cd ξ (x, px) + c. (9)
Now, by the bounds on curvature -b 2 ≤ K X ≤ -1 we get (cp. [START_REF] Heintze | Geometry of horospheres[END_REF])

2 sinh d(x, px) 2 ≤ d ξ (x, px) ≤ 2 b sinh b 2 d(x, px) hence d(x, px)/d(x 0 , px 0 ) C ′
≍ 1 for a constant C ′ > 0 only depending on b and c.2 Proposition 3.5 Let α and β be two hyperbolic isometries in Γ with, respectively, repelling and attractive fixed points α -, α + , β -, β + ). Then

lim n→+∞ e l(α n )+l(β n )-l(β n α n ) = [α -, β -, α + , β + ]
where l(γ) denotes the length of the closed geodesic corresponding to γ for any γ ∈ Γ.

The set of couples (α -, α + ) of all hyperbolic fixed points of Γ being dense in

X(∞) 2 , this shows that [f ∞ (ξ 1 ), f ∞ (ξ 2 ), f ∞ (ξ 3 ), f ∞ (ξ 4 )] = [ξ 1 , ξ 2 , ξ 3 , ξ 4 ] ∀ξ 1 , ξ 2 , ξ 3 , ξ 4 ∈ X(∞).
Proof of Proposition 3.5. Fix x ∈ X. For n ≥ 0, set γ n = β n α n , and let γ - n , γ + n be its repelling and attractive fixed points. Consider two sequences of points

a k ∈ ]α -, α + [ and b k ∈ ]β -, β + [ such that lim k→+∞ a k = α -and lim k→+∞ b k = β + ; moreover, we can choose a sequence n k → ∞ such that lim k→+∞ α n k a k = α + and lim k→+∞ β -n k b k = β -.
Now, for each k, let B k be a compact ball centered at x containing a k and b k . Notice that γ - n and γ + n tend respectively to α -and β + , so the sequence of geodesics ]γ - n , γ + n [ tend to ]α -, β + [: namely, for k fixed, the distance between ]γ - n , γ + n [ and ]α -, β + [, restricted to the compact set B k , tends to 0 when n → ∞. We can then choose n k large enough so that

d ]γ - n k , γ + n k [ ∩B k , ]α -, β + [ ∩B k < 1/k Call a ′ k , b ′ k the projections of a k , b k on ]γ - n k , γ + n k [; so, the sequences (a ′ k ) k and (b ′ k )
k also converge respectively to α -and β + , and the sequences (α n k a ′ k ) k , (β -n k b ′ k ) k respectively to α + and β -. We then have:

[α -, β -, α + , β + ] = lim k→∞ e d(a ′ k ,α n k a ′ k )+d(β -n k b ′ k ,b ′ k ) e d(a ′ k ,b ′ k )+d(β -n k b ′ k ,α n k a ′ k ) = lim k→∞ e d(a k ,α n k a k )+d(β -n k b k ,b k ) e d(a ′ k ,b ′ k )+d(β -n k b ′ k ,α n k a ′ k ) (10)
by definition of the cross-ratio. Notice that the numerator gives exacty e l(α n k )+l(β n k ) , as the points a k and b k lie on the axes of α, β respectively. On the other hand, for

k ≫ 0 d(a ′ k , b ′ k ) + d(β -n k b ′ k , α n k a ′ k ) = l(γ n k ). ( 11
)
Indeed, let V β (b k ) be the hyperplane orthogonal to the axis of β, passing through b k .

When k is large enough, the point α n k a ′ k is close to α + , in particular it belongs to the half space bounded by

β -n k (V β (b k )) which contains β + ; consequently, the point γ n k a ′ k = β n k α n k a ′ k belongs to the half-space V β (b k ) which contains β + , so b ′ k lies on the geodesic (γ - n k , γ + n k ) between a ′ k and γ n k a ′ k . As d(b ′ k , γ n k a ′ k ) = d(β -n k b ′ k , α n k a ′ k )
, the equality (11) readily follows. Letting k → +∞ in [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF] then achieves the proof.2

Entropy rigidity

This section is devoted to the proof of Theorem 1.2. The proof is through the method of barycenter, initiated by Besson-Courtois-Gallot [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF], [START_REF] Besson | Lemme de Schwarz réel et applications géométriques[END_REF], and follows the lines of [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF] (Theorem 1.6, holding for compact quotients). The main difficulty in the finite volume, non compact case is to show that the map produced by the barycenter method is proper: we will recall in §4.2 the main steps of the construction, referring the reader to [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF] for the estimates which are now well established, while we will focus on the new estimates necessary to prove properness. For this, we will need accurates estimates of the Patterson-Sullivan measure of some subsets of X(∞), which we will describe in the first subsection.

On the Patterson measure of non uniform lattices

The Patterson-Sullivan measures of Γ are a family of finite measures µ = (µ x ), indexed by points of X and with support included in the limit set LΓ ⊂ X(∞), satisfying the following properties (cp. for instance [START_REF] Sullivan | The density at infinity of a discrete group of hyperbolic motions[END_REF], [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] for details about their construction):

1. they are absolutely continuous w.r. to each other: for any

x, x ′ ∈ X dµ x ′ dµ x (ξ) = e -δ Γ b ξ (x ′ ,x) (12)
2. they are Γ-equivariant: for every γ ∈ Γ and every Borel set A ⊂ X(∞)

µ x (γ -1 A) = µ γx (A) ( 13
)
When Γ is a lattice, we will use the decomposition of X explained in §2.1 to describe the local behavior of Patterson-Sullivan measures of Γ on the limit set Λ Γ = X(∞). The proposition below gives an uniform estimate, which will be crucial in the sequel, for the measure µ x (V ζ (x, R)) of "small" spherical caps, i.e. when R ≫ 0.

So, let D = K ∪ C 1 ∪ • • • ∪ C l
be a decomposition of the Dirichlet domain of Γ centered at some fixed point x, corresponding to the maximal, bounded parabolic subgroups P 1 , ..., P l of Γ with fixed points ξ 1 , ..., ξ l as described in 2.1. If xζ(R) projects to the thick part K of X, then formulas ( 12) and ( 13) easily give the uniform lower estimate:

µ x (V ζ (x, R)) c e -δ Γ R (14)
(where c is a positive constant, depending on the minimal mass of a spherical cap at distance less than D = diam(K) from x). On the other hand, when xζ(R) projects to the cuspidal part, we have: Proposition 4.1 There exists a constant c > 0 satisfying the following property. Let ζ ∈ X(∞) and assume that the point xζ(R) belongs to γC i , R > 0. Then:

µ x (V ζ (x, R)) c e -δ Γ (R+r) v P i (x, 2r) (15)
where

r = b ξ i (x, γ -1 xζ(R)).
This estimate stems from a series of technical lemmas, and might be deduced from work developed in [START_REF] Peigné | Autour de l'exposant de Poincaré d'un groupe kleinien[END_REF] and [START_REF] Schapira | Lemme de l'ombre et non divergence des horosphères d'une variété géométriquement finie[END_REF] (notice however that, in [START_REF] Peigné | Autour de l'exposant de Poincaré d'un groupe kleinien[END_REF], µ x has no atomic part, and in [START_REF] Schapira | Lemme de l'ombre et non divergence des horosphères d'une variété géométriquement finie[END_REF] the parabolic subgroups are assumed to satisfy an additional, strong regularity assumption). Since the estimate is of independent interest, we will report for completeness the proof of Proposition 4.1, in full generality, in the Appendix.

Entropy rigidity : proof of Theorem 1.2

Let X = Γ\X, fix a point x 0 ∈ X and call for short b ξ (x) = b ξ (x, x 0 ). The function b ξ is strictly convex if K X ≤ -1 < 0, since for every point y we have:

Hess y b ξ ≥ g y -(db ξ ) y ⊗ (db ξ ) y (16)
where g denotes the metric tensor of X; moreover, it is known that if the equality holds in ( 16) at every point y and for every direction ξ, then the sectional curvature is constant, and X is isometric to the hyperbolic space H n . The idea of the proof is to show that the condition δ Γ = n -1 forces the equality in [START_REF] Hamenstädt | Entropy-rigidity of locally symmetric spaces of negative curvature[END_REF].

Recall that, for every measure µ on X(∞) whose support is not reduced to one point, we can consider its barycenter, denoted bar[µ], that is the unique point of minimum of the function y → B µ (y) = X(∞) e b ξ (y) dµ(ξ) (notice that this is C 2 and strictly convex function, as b ξ (y) is). If supp(µ) is not a single point, it is easy to see that lim y→ξ B µ (y) = +∞ for all ξ ∈ X(∞) cp. [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF].

Consider now the map F : X → X defined by

F (x) = bar e -b ξ (x) µ x = argmin y → X(∞) e b ξ (y,x) dµ x (ξ)
where (µ x ) x is the family of Patterson-Sullivan measures associated with the lattice Γ.

In [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF] it is proved that the map F satisfies the following properties:

(a) F is equivariant with respect to the action of Γ, i.e. F (γx) = γF (x); (b) F is C 2 , with Jacobian:

|Jac x F | ≤ δ Γ + 1 n n det -1 (k x ) (17)
where k x (u, v) is the bilinear form on T x X defined as

k x (u, v) = X(∞) e b ξ (F (x),x)) (db ξ ) 2 F (x) + Hess F (x) b ξ (u, v) dµ x (ξ) X(∞) e b ξ (F (x),x)) dµ x (ξ) (18)
Notice that the eigenvalues of k x are all greater or equal than 1, by [START_REF] Hamenstädt | Entropy-rigidity of locally symmetric spaces of negative curvature[END_REF]. Property (a) stems from the equivariance (i) of the family of Patterson-Sullivan measures with respect to the action of Γ, and from the cocycle formula for the Busemann function: b ξ (x 0 , x) + b ξ (x, y) = b ξ (x 0 , y). Property (b) comes from the fact that the Busemann function is C 2 on Hadamard manifolds, and is proved by direct computation, which does not use cocompactness. By equivariance, the map F defines a quotient map F : X → X, which is homotopic to the identity through the homotopy

Ft (x) = bar e -b ξ (x) (tµ x + (1 -t)λ x ) mod Γ, t ∈ [0, 1]
where λ x is the visual measure from x (with total mass equal to the volume of S n-1 ).

Actually, the map F t = bar e -b ξ (x) (tµ x + (1t)λ x ) defines a map between the quotient manifolds, as it is still Γ-equivariant; moreover, we have bar e -b ξ (x) λ x = x since, for all v ∈ T x X:

dB e -b ξ (x) λx x (v) = X(∞) (db ξ ) x (v)e b ξ (x) e -b ξ (x) dλ x (ξ) = UxX g x (u, v)du = 0.
We will now prove that:

Proposition 4.2
The homotopy map Ft is proper.

Assuming for a moment Proposition 4.2, the proof of Theorem 1.2 follows by the degree formula: since F is properly homotopic to the identity, it has degree one, so vol

( X) = X F * dv g ≤ X |Jac x F |dv g ≤ δ Γ + 1 n n X det -1 (k x )dv g ≤ δ Γ + 1 δ Γ (H n ) + 1 n vol( X)
as det(k x ) ≥ 1 everywhere. So, if δ Γ = δ Γ (H n ) = n -1, we deduce that det(k x ) = 1 everywhere and k = g, hence the equality in the equation ( 16) holds for every y = F (x) and ξ. Since F is surjective, this shows that X has constant curvature -1.2

Proof of Proposition 4.2. Denote by z the projection of a point z ∈ X to X, and set δ = δ Γ ; recall that δ = n -1, but we will use this property only at the end of the proof. Let µ t

x = e -b ξ (x) (tµ x + (1t)λ x ): we need to show that if t k → t 0 and if xk → ∞ in X, then ȳk = Ft k (x k ) = bar[µ t k

x k ] goes to infinity too. Now, assume by contradiction that the points ȳk stay in a compact subset of X: so (up to a subsequence) xk , ȳk lift to points x k , y k such that y k → y 0 ∈ X and (ξ) = e -δb ξ (x,y 0 ) , we have

d(y 0 , x k ) = d(ȳ 0 , xk ) = R k → ∞.
(dB µ t x ) y (v) = t X(∞)
(db ξ ) y (v) e b ξ (y,y 0 ) e (δ+1)b ξ (y 0 ,x) dµ y 0 (ξ)

+(1 -t) X(∞)
(db ξ ) y (v)e b ξ (y,x) dλ x (ξ) [START_REF] Heintze | Geometry of horospheres[END_REF] We will now estimate the two terms in [START_REF] Heintze | Geometry of horospheres[END_REF] and show that (dB

µ t k x k
) y k does not vanish for R k ≫ 0, a contradiction. So, let ζ k be the endpoints of the geodesic rays y 0 x k and let

v k = (∇b ζ k ) y k . Also, consider the spherical caps V ζ k (y 0 , R k /2) and V ζ k (y 0 , R k ).
Let us first consider the contributions of the two integrals of the right hand side in [START_REF] Heintze | Geometry of horospheres[END_REF] over

X V ζ k (y 0 , R k /2). If ξ ∈ X(∞) V ζ k (y 0 , R k /2
), the projection of ξ over y 0 ζ k falls closer to y 0 than to x k , hence b ξ (y 0 , x k ) ≤ 0; moreover, |b ξ (y k , y 0 )| ≤ d(y k , y 0 ) → 0, so the first integral on X V ζ k (y 0 , R k /2) for x = x k , y = y k and v = v k gives:

X V ζ k (y 0 , R k 2 ) (db ξ ) y k (v k )e b ξ (y k ,y 0 ) e (δ+1)b ξ (y 0 ,x k ) dµ y 0 ≤ 2 µ y 0 for k ≫ 0. Analogously, the second integral on X V ζ k (y 0 , R k /2) yields X V ζ k (y 0 , R k 2 ) (db ξ ) y k (v k )e b ξ (y k ,x k ) dλ x k ≤ 2 vol(S n-1 )
for k ≫ 0, since |b ξ (y k , x k )-b ξ (y 0 , x k )| ≤ d(y k , y 0 ). So, these contributions are bounded. We now compute the contributions of the integrals over

V ζ k (y 0 , R k /2) V ζ k (y 0 , R k ). For all ξ ∈ V ζ k (y 0 , R k /2) we have that (∇b ξ ) y 0 (∇b ζ k ) y 0 is close to 1, for R k ≫ 0; moreover, as |(∇b ξ ) y k v k -(∇b ξ ) y 0 (∇b ζ k ) y 0 | k→∞ -→ 0, we deduce that (db ξ ) y k (v k ) > 1 2 on V ζ k (y 0 , R k /2
) for k ≫ 0, hence these contributions are positive.

Finally, let us compute the contributions of these integrals on the caps V ζ k (y 0 , R k ). For ξ ∈ V ζ k (y 0 , R k ), consider the ray y 0 ξ from y 0 to ξ, and the projection P (t) on the geodesic y 0 ζ k of the point ξ(t) := y 0 ξ(t). We have, by [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] b

ξ (y 0 , x k ) ≥ lim t→∞ [d(y 0 , P (t)) + d(P (t), ξ(t))] -[d(ξ(t), P (t)) + d(P (t), x k )] -ǫ0 ≥ R k -ǫ with ǫ = ǫ(π/2
). Therefore we deduce that, for k ≫ 0, we have

V ζ k (y 0 ,R k )) (db ξ ) y k (v k )e b ξ (y k ,y 0 ) e (δ+1)b ξ (y 0 ,x k ) dµ y 0 ≥ 1 4 e (δ+1)(R k -ǫ) µ y 0 (V ζ k (y 0 , R k )) (20) V ζ k (y 0 ,R k )) (db ξ ) y k (v k )e b ξ (y k ,x k ) dλ x k ≥ 1 4 e (R k -ǫ) vol(S n-1 ). ( 21
)
It is clear that the right-hand side of (21) goes to infinity when R k ≫ 0; we will now prove that the right-hand side of (20) also diverges for R k → ∞. This will conclude the proof, as it will show that dB µ t k

x k (v k ) does not vanish for k ≫ 0 (being a convex combination of two positively diverging terms). So, let D = K ∪ C 1 ∪ • • • ∪ C l be a decomposition of the Dirichlet domain of Γ centered at y 0 as in (2), corresponding to maximal, bounded parabolic subgroups P 1 , ..., P l with fixed points ξ 1 , ..., ξ l . We know that xk belongs to some cusp of X, so x k ∈ γC i for some γ; let then

r k = b ξ i (y 0 , γ -1 x k ) ≤ R k . If γξ i falls in V ζ k (y 0 , R k ) and δ ≫ 0, as K X ≥ -
e (δ+1)R k µ y 0 (V ζ k (y 0 , R k )) e (δ+1)R k e -δ(R k +r k ) n≥0 v P i (2r)e -δn e R k -δr k v P i (2r k ) e R k -δr k A P i (x 0 , y 0 , r k )
.

Since K X ≤ -1, we know that A P i (x 0 , y 0 , r k ) e -(n-1)r k , so we obtain

e (δ+1)R k µ y 0 (V ζ k (y 0 , R k )) e R k +(n-1-δ)r k .
On the other hand, when γξ i ∈ V ζ k (y 0 , R k ), we have, by Propositions 4.1 and 2.3:

e (δ+1)R k µ y 0 (V ζ k (y 0 , R k )) e R k -δr k v P (y 0 , 2r k ) e R k +(n-1-δ)r k .
Both lower bounds tend to +∞ as k → +∞, since R k → +∞ and δ ≤ n -1; thus, the integral in [START_REF] Katok | Entropy and closed geodesics[END_REF] diverges. This concludes the proof that the map Ft is proper.2

Appendix

We report here, for completeness, a proof of the estimate given in Proposition 4.1.

To prove Proposition 4.1, we will need a series of elementary lemmas, where some equalities hold up to some constant: so we will use the symbol f C ≈ g to mean that two quantities f and g differ of at most C. To avoid cumbersome notations, we will give the same name to these constants in all the lemmas, meaning that they all hold for the choice of a suitable constant C large enough. All these constants will depend on the upper bound of the sectional curvature K X ≤ -1 and, possibly, on other parameters of X = Γ\X which we will specify case by case.

Recall that a parabolic group P of isometries fixing ξ ∈ X(∞) is called bounded if it acts cocompactly on X(∞) -{ξ} (as well as on every horosphere ∂H centered at ξ). If D(P, x) is the Dirichlet domain of P centered at x, the sets S x = D(P, x) ∩ ∂H ξ (x) and the trace at infinity S x (∞) = D(P, x)∩X(∞) of D(P, x) are compact, fundamental domains for the action of P on ∂H ξ (x) and on X(∞), respectively.

It is thus sufficient to show that

µ z R V ζ (x, R)
e -δ Γ r v P (x, 2r) [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF] For this, we will analyse two different cases:

µ x R (V ξ (x, R)) ≥ µ x R ({ξ}) + p∈P pS(∞)⊂V ξ (x,R) µ x R (pS(∞)) ≥ p∈P d(x,px)≥2R+C µ x R (pS(∞)). ( 23
)
From the equivariance and the density formula ( 12), ( 13) for the family µ x we get

µ x R (pS(∞)) = S(∞)
e -δ Γ bpη(x R ,px R ) µ x R (dη) ≍ µ x R (S(∞))e δ Γ (2R-d(x,px))

because b pη (x R , px R ) ≈ d(x, px) -2R, by Lemma 5.3. Now, µ x R (S(∞)) ≍ e -δ Γ R µ x (S(∞)) ≍ e -δ Γ R , since X(∞) = P S(∞) ∪ {ξ} and the mass of µ x R is not reduced to one atom, so µ x (S(∞)) > 0. Therefore, from [START_REF] Knieper | On the asymptotic geometry of nonpositively curved manifolds[END_REF] we deduce that, for ∆ large enough, we have

µ x R (V ξ (x, R)) e δ Γ R p∈P i 2R+C+∆≥d(x,px)≥2R+C
e -δ Γ d(x,px) e -δ Γ R v P (x, 2R) [START_REF] Nguyen Phan | On finite volume, negatively curved manifolds[END_REF] as v ∆ P (x, 2R) ≍ v P (x, 2R) by Proposition 2.3, if ∆ ≥ ∆ 0 . The estimate [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF] follows in this case, since ζ = ξ and z R = x R , so r = b ξ (x, z R ) = b ξ (x, x R ) = R. b) Assume now that ζ = γξ for some γ = 1. We then set ξ ′ = γξ = ζ, x ′ = γx, H ξ ′ = γH ξ , x ′ t = γx t and R ′ := b ξ ′ (x ′ , z R ). Notice that, without loss of generality, we can assume that x ′ lies at distance less than diam(K) from the geodesic ray xξ ′ (actually, as P acts cocompactly on ∂H ξ , we can replace γ by γp for some suitable p ∈ P ), so d(z R-R ′ , x ′ ) and d(z R , x ′ R ′ ) are both bounded by 2diam(K). By Lemma 5.2(i), there exists C > 0 such that

V ξ ′ (x, R) = V ξ ′ (z R-R ′ , R ′ ) ⊃ V ξ ′ (x ′ , R ′ + C)
and then [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF] follows from a), by applying the inequality (24) to ξ ′ , x ′ and R ′ . Actually, as d(z R , x ′ R ′ ) is bounded, we have dµ x ′ R ′ /dµ z R ≍ 1 and we get from (24)

µ z R (V ξ ′ (x, R)) µ x ′ R ′ (V ξ ′ (x ′ , R ′ + C
)) e -δ Γ R ′ v P (x ′ , 2R ′ ) which gives [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF], since in this case ξ ′ = ζ and r = b ξ (x, γ -1 z R ) = b ξ ′ (x ′ , z R ) = R ′ . proceed as above, according to the cases ξ ′ ∈ V ζ (x, R -2C) or ξ ′ ∈ V ζ (x, R -2C). In the first case, we call S := R -4C, S ′ = b ξ ′ (x ′ , z S ), so V ζ (x, S) ⊃ V ξ ′ (x, S + C) and we have d(z R , x ′ S ′ ) ≤ 6C + 2diam(K); then, using Lemma 5.2, we deduce similarly to [START_REF] Otal | Le spectre marqué des longueurs des surfaces à courbure négative[END_REF], that

µ z R V ζ (x, S) µ x ′ S ′ V ξ ′ (x ′ , S ′ + 2C) e -δ Γ S ′ v P (x ′ , 2S ′ )
which yields [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF], as here r = b ξ (x, γ -1 z R ) ≈ b ξ ′ (x ′ , z S ) = S ′ . In the second case, we call again ζ the point at infinity opposite to ζ with respect to x, x the point of ]ζ, x] ∩ ∂H ξ ′ closest to ζ, and we set R := d(x, z R ), x R := xξ ′ ( R). So, z R = xζ ( R), d(z R , x R) < 2C and dµ x R /dµ z R ≍ 1. Then, we deduce as before that V ζ (x, R) ⊃ X(∞) V ξ ′ (x, R + C) and we obtain, analogously to [START_REF] Peigné | Autour de l'exposant de Poincaré d'un groupe kleinien[END_REF], that

µ z R (V ζ (x, R)) e -δ Γ R v P (x, 2 R)
which concludes the proof as, in this case, R = b ξ ′ (x, x R) ≈ b ξ ′ (x ′ , z R ) = r .2

  [x, y] (resp. [x, ξ[) the geodesic segment from x to y (resp. the ray from x to ξ), and by xξ(t) the parametrization of geodesic ray [x, ξ[ by arc length. Let b ξ (x, y) = lim z→ξ d(x, z)d(z, y) be the Busemann function centered at ξ; the level set ∂H ξ (x) = {y | b ξ (x, y)| = 0} (resp. the suplevel set H ξ (x) = {y | b ξ (x, y) ≥ 0} is the horosphere (resp. the horoball)

For

  x ∈ X and ζ ∈ X(∞), we consider the point xζ(R) at distance R from x on the geodesic ray [x, ζ[, and define the "spherical cap" V ζ (x, R) ⊂ X(∞) as the set of points of X(∞) whose projection on the geodesic ray [x, ζ[ falls between xζ(R) and ζ.

  By the cocycle relation b ξ (y, x) = b ξ (y, y 0 ) + b ξ (y 0 , x) and by the density formula for the Patterson-Sullivan measures dµx dµy 0

Case 1 :

 1 ζ ∈ Γξ. a) Assume first γ = 1, so ζ = ξ and z R = x R ∈ H ξ .We have, byLemma 5.3:

  b 2 we can use Propositions 4.1 and 2.3 to deduce that

Cp. Lemma 2.4 in[START_REF] Hamenstädt | Time preserving conjugacies of geodesic flows[END_REF], which is central in the argument: it is based on the fact that the closed geodesics on X equidistribute towards the Bowen-Margulis measure when X is compact. This property does not hold for non uniform lattices as soon as the Bowen-Margulis measure is infinite.

Cp. the definition of the maps Ψ b c in[START_REF] Storm | The minimal entropy conjecture for nonuniform rank one lattices[END_REF], which clearly require that c is greater than the exponential growth rate of the universal covering of the manifold under consideration.

This seems unclear even under the assumption of a C 1 -conjugacy; cp. the proof of Proposition 1.2 in[START_REF] Croke | Conjugacy and rigidity for manifolds with a parallel vector field[END_REF], where Stokes's theorem fails, unless one knows that the conjugacy F has bounded derivatives.

For instance, Corollary 4.2 in[START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF] only holds for µH -integrable functions, and cannot be applied as it is to constant functions or to tr U + (v) to deduce Theorem 5.1, when µH = ∞.

The following Lemmas can be found, for instance, in [START_REF] Schapira | Lemme de l'ombre et non divergence des horosphères d'une variété géométriquement finie[END_REF] (Lemmes 2.6, 2.7 and 2.9): Lemma 5.1 There exists a constant C > 0 with the following property. Let x ∈ X and ζ ∈ X(∞) be fixed. Then, for any ξ ∈ V ζ (x, R) we have:

Lemma 5.2 There exists a constant C > 0 with the following property. Let x ∈ X and ζ ∈ X(∞) be fixed. Then: (i) for any x ′ such that d(x, x ′ ) < C we have

Lemma 5.3 Let P a bounded parabolic subgroup of X fixing ξ, and let S x (∞) as above. There exists a constant C > 0 (depending on the diameter of S x ) with the following properties: for any p ∈ P

be a decomposition of the Dirichlet domain of Γ centered at x, corresponding to the maximal, bounded parabolic subgroups P 1 , ..., P l of Γ with parabolic fixed points ξ 1 , ..., ξ l , and with C i = D ∩ H ξ i , as described in 2.1. Moreover, let S i (∞) = D∩X(∞) be the fundamental domains for the action of P i on X(∞)\{ξ i }, and let z t := xζ(t) and x i,t := xξ i (t). We assume that z R belongs to γH ξ i ; so, call for short ξ = ξ i ,

2 ) as in [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]. Thus, the density formula (12) yields

for some constant c > 0 only depending on the upper bound of the curvature.

Notice that we have d(z R , x S ) ≤ 5C by Lemma 5.1. Therefore, applying again (24) to ξ, x and S, we get

and the estimate ( 22) follows, since here

On the other hand, if

Hence,

Similarly to case 1, we can estimate this by applying Lemma 5.3 to x and ξ :

as

so, combining [START_REF] Otal | Sur la géométrie symlectique de l'èspace des géodésiques d'une variété à courbure négative[END_REF] and ( 27) we obtain

)) e -δ Γ R v P (x, 2 R) e -δ Γ r v P (x, 2r)

(since x is at bounded distance from the orbit of x).

b) Assume now that γ = 1. We set ξ ′ = γξ, x ′ = γx, H ξ ′ = γH ξ , x ′ R = γx R , with d(x ′ , [x, ξ[) ≤ diam(K), and we