Niharika Yadav
email: y.niharika27@gmail.com

Vinay Kumar

A REVIEW OF VARIOUS TEXT LOCALIZATION TECHNIQUES

Keywords: Scene images, Text localization, CC-based method, Texture based method

Text information extraction is a growing area of research. Enormous work has been done to efficiently and robustly extract the text regions from scene text images. Various Text extractions models have been defined comprising of various stages. Amongst them text localization is an important stage and thus researchers mainly focus on this stage. This paper provides a review of various techniques of text localization. The paper also includes a comparison of the performance of these techniques on the parameters such as precision rate and recall rate.

I. INTRODUCTION

Text extraction is an arduous task in document image Analysis. Text in images contains information which is noteworthy and can be utilised for structuring of images or indexing databases. This also has its usefulness for many real world applications, such as navigation, licence plate detection, conversion of paper based collection to e-books using OCR's etc. [START_REF] Jung | Text information extraction in images and video: a survey[END_REF].text embedded in the mages contains a systematic information which can be used as a content based access to digital images, if it is extracted and harnessed efficiently.

Text appearing in images is broadly classified in three categories: Document text, Scene text, Digital-born text [START_REF] Wang | End-to-end scene text recognition[END_REF]. Amongst these extracting texts from the scene text images is most challenging as this type of text can have variable orientations, font size etc. In addition they may also be distorted due to the perceptive projections and may have low contrasts. Figure 1 shows examples of scene text images.

A generalised Text Information Extraction (TIE) model has been shown in Figure 2.

Regardless of the profound research to develop an efficient model there are various problems with a scene text image that make automatic text extraction a challenging task.

A. Region Based Approach:

Region-based method uses the properties of the color or gray scale in the text region or their differences to the corresponding properties of the background [START_REF] Sawaki | Automatic acquisition of context-based images templates for degraded character recognition in scene images[END_REF]. They utilise the property that the variation of color within text is minimal moreover there is a sufficient distinction between the text and the background. Text can be obtained by thresholding the image at intensity level in between the text color and that of its immediate background. This approach is implemented in bottom-up fashion. Sub structures are indentified and merged to form homogeneous region which are marked with bounding boxes. This method is further divided into two sub-approaches:

Connected component based (CC-based):

Small similar components of the input images are grouped to form successively large components using bottom up approach in the case of CC-based methods. This process iteratively continues until all the regions of the image are not identified. Filtering of the non text regions of the image is performed by considering a geometrical analysis which merges the components using the spatial arrangement of the components and mark the boundaries of the text regions.

Edge Based Methods:

Edges are those portions of the image that corresponds to the object boundaries. There is a major contrast in the text and the background and this drastic variation forms the edges.

Text localization

Region based methods

CC-Based method

Edge Based method

Texture based methods

Hybrid Methods

Morphlogical operations

The non text region of the images is filtered out by firstly identifying the edges of the text boundaries and then applying various heuristics [21]. Generally, an edge filter is incorporated for the identification of the edges in an image, and for the merging stage an averaging operation or a morphological operator is used.

B. Texture Based Approach

Texture-based methods exploit textural features of the text. These properties distinguished the text from its background [START_REF] Gao | An adaptive algorithm for text detection from natural scenes[END_REF]. Parameters such as energy, contrast, correlation, and entropy define the texture of an image. Texture based methods implement techniques such as FFT, wavelet spatial variance etc.

C. Morphological Approach

Mathematical morphology is a topological and geometrical based approach for image analysis [START_REF] Sternberg | Grayscale morphology[END_REF].

Character recognition and document analysis uses the morphological operations because operations like translation, rotation and scaling do not have any effect on the geometrical shape of the image.

This method works robustly under different image alterations.

A detailed study of various techniques based on the above approach has been done and review is given in the next section.

III. LITERATURE REVIEW

This section provides a detailed review of various localization techniques.

A. Connected Component Based Method

Sobattka et al. [START_REF] Sobottka | Text extraction from colored book and journal covers[END_REF] proposed a CC-based method that implemented bottom up and top down analysis on images of book covers and journals. Histogram based clustering was implemented in the preprocessing step which reduced the color variation in the image significantly. For the text localization step, in bottom-up analysis homogenous regions were identified while in case of bottom-up analysis image was iteratively split horizontally and vertically. Finally the two results were compared to obtain the text regions of the image.

The method gave a character recognition rate of 98%.

Hase et al. [START_REF] Hase | Character String Extraction from Color Documents[END_REF] proposed a CC-based method for color documents. A common assumption of the machine printed characters is that each character is printed with a single color. Color histogram was generated by converting the Pixel values to a new color space. In the next step the image is divided in various binary images and a multi-stage relaxation technique is applied to extract strings .In the next step all the results obtained from individual images were merged and character strings were identified using likelihood using conflict resolution rules.

Jain and Yu [START_REF] Jain | Automatic text location in images and video frames[END_REF]

B. Texture Based Method:

Shiva kumara et.al [START_REF] Liu | Multiscale edge-based text extraction from complex images[END_REF] proposed a new method for detection of handwritten text from videos. The algorithm was divided in five steps. In the first stage. Averaging of the RGB planes was performed to enhance the text areas of image. Maximum Color Difference (MCD) criteria then applied to restore the contrast of the image and to highlight the test regions in the next stage, to cluster the text regions K-mean clustering was performed with K =2. In the following stage the candidate region were bounded using sobel filter. In the next stage, a novel Boundary Growing Method (BGM) based on the nearest neighbour concept was used to adjust the boundary of the text. The method fixed the boundaries by considering the fact that in one direction there is regular spacing between words and in the final stage, the false positive edges obtained due to BGM were eliminated using the concept of intrinsic and extrinsic edges.

Shyama et.al [START_REF] Shivakumara | A New Method For Handwritten Scene Text Detection In Video[END_REF] proposed a text segmentation technique that extracted text by projection algorithm. Color based segmentation was performed to link consecutive pixels that lied in the same direction. Light Edge Enhancement (LEE) was used to locate the regions having consecutive pixel and to enhance the edges. Next, the motion blur was removed with the help of heavy edge enhancement (HEE) from camera image Kim et al. [START_REF] Kim | Texture-based approach for text detection in images using support vector machines and continuously adaptive mean shift algorithm[END_REF] proposed a method that could detect texts using texture properties. A support vector machine (SVM) was implemented to study the properties of textural basis of the image. The method used the raw pixels of the image that confided with the textural pattern rather than adding new external texture feature extraction model. This worked fine even for the high dimensional spaces. Next step of the method was to implement, continuously adaptive mean shift algorithm (CAMSHIFT) that could locate the text regions by using the analysis of the texture properties. The combination of CAMSHIFT and SVMs produced very resourceful results as text detection was significant. The performance of the system was based on the criterion that the SVM was able to classify the text and non text regions and not on the final text detection. The author used a set of 100 images that were divided in two sets. The first set comprised of 70 images that were used for training and the other 30 were used to validate the results obtained.

Edge Based Methods:

Xiaoqing Liu [START_REF] Chowdhury | Robust Extraction of Text from Camera Images using Colour and Spatial Information Simultaneously[END_REF] Chen et al. [38] implemented canny edge operator that identified the high contrast regions of the image. In order to reduce the computational complexity a single edge point was used in one window for the estimation and orientation. The information obtained is then used to enhance the image by performing morphological operations on images that could connect the edges into clusters. Morphological operations were performed on the image to connect the edges into clusters. Two types of Gabor asymmetric filters were implemented to generate a general estimation: an edge-form filter and a stripe-form filter. The obtained information about the edge is then enhanced using a suitable scale. This resulted in the blurring of the area that had no specific scale. Image enhancement was then obtained using the localization techniques.

Jain and Chugh [START_REF] Jain | Charecter Localization From Natural Images Using Nearest Neighbours Approach[END_REF] proposed a method based on the nearest neighbour approach. The algorithm has four stages i) Color Processing ii) Detection of Edges iii) Text Area Localization iv) Gap Filling and Enhancement. In the preprocessing step color processing is performed by first converting the image to YUV format. In the next stage an edge image is generated to highlight the text regions of the image. To localize the text areas, height-wise and width-wise analysis is done through the horizontal and vertical histogram for the target text areas. A geometric proportion between the length and width is modelled experimentally to remove the probable non-text regions. Areas with ratio of less than ten are marked text regions. The algorithm was evaluated for a dataset of 25 images with variation in font size, perspective and alignment under different lightning conditions.

C. Morphological Based Method

Jui-Chen Wu [START_REF] Wu | Morphology-based text line extraction[END_REF] proposed a method that consists of three stages a) feature extraction, b)

Text localization, and c) Text verification. In the first stage, the high contrast feature of the text region was extracted using a novel set of morphological operations. In the second stage, the possible text regions were marked using a labelling technique. The skewness of the text lines was handled using a moment based method. The potential text segments were extracted using a rule based scheme. Furthermore the text lines were merged by defining similarity measures. In the final stage, an x-projection technique was proposed that was able to extract all the regularities from the text lines.

Luz et.al [START_REF] Alves | Text Regions Extracted from Scene Images by Ultimate Attribute Opening and Decision Tree Classification[END_REF] implemented morphological filters to extract the candidate text regions from the scene images. Closing and opening operations were performed on the connected components. A decision tree classifier was used to differentiate between the text and non text regions.

IV. COMPARISON AND PERFORMANCE ANALYSIS

Analysis of the various techniques for text localization is shown in Table I. In order to calculate the robustness and performances of the proposed algorithms two methods are defined. The Precision rate is defined as the ratio of correctly detected words to the sum of correctly detected words plus false positives.

(

) 1
The Recall rate is defined as the ratio of correctly detected words to the sum of correctly detected words plus false negatives.

(

) 2
Table II shows the performance analysis of the text localization techniques.

Table I Comparison of various techniques Technique

Figure 1 :

 1 Figure 1: Examples: Scene text images

Figure 2 :

 2 Figure 2: Text Information Extraction Model

Figure 3 :

 3 Figure 3: Classification: Text Localization Techniques

  applied a model that included following stages. In the first stage, preprocessing, this included bit dropping. A 24-bit image was bit dropped to 6-bit image. In the second stage of color clustering, image was quantized after the decomposition of the image in multiple foregrounds; same localization technique is implied on each foreground image. The authors used a block adjacency graph to generate the CCs for each foreground image. The localized text components of all the individual foreground images were then merged to obtain an output image. The algorithm gave promising results for horizontal and vertical texts however text detection was poor in case of skewed text.

  method consists of three stages: candidate text region detection, text region localization and character extraction. The first stage implements a multiscale edge detector where multiscale images are produced by Gaussian low pass filter which undergo down sampling to reduce the image in both horizontal and vertical direction. A feature map is then generated that separates the texture like regions from text regions. In the second stage text localization is performed implying clustering techniques. In the third stage, already proposed existing OCR engine where used, that were only efficient with printed documents having plane background and are unable to give promising results in case of text embedded in the image or in case of complex backgrounds.

Author Approach Remark

CC-Based

Approach Sobattka et al. [START_REF] Sobottka | Text extraction from colored book and journal covers[END_REF] Implemented bottom-up and top down analysis Robust to size of text Hase et al. [START_REF] Hase | Character String Extraction from Color Documents[END_REF] Implemented likelihoods using conflict resolution rules.

Works for curved and shadowed strings.

Jain and

Yu [START_REF] Jain | Automatic text location in images and video frames[END_REF] Implemented Bit dropping Poor performance when color variance is less. Liu [START_REF] Chowdhury | Robust Extraction of Text from Camera Images using Colour and Spatial Information Simultaneously[END_REF] Based on multiscale images using Gaussian pyramids.

Robust and effective for complex images

Chen et al. [START_REF] Chen | Text enhancement with asymmetric filter for video OCR[END_REF] Based on canny edge detector Blurs or eliminates regions that do not have specified spaces.

Jain and

Chugh [START_REF] Jain | Charecter Localization From Natural Images Using Nearest Neighbours Approach[END_REF] Based Luz et.al [START_REF] Wu | Morphology-based text line extraction[END_REF] 86.9 85

V. CONCLUSION

Content retrieval from images for various proposes has been performed for many years. This paper provides a broad study of various techniques propounded over the years of research.

The paper presents a comparison of approaches that utilize diverse features of scene text images. A performance analysis of all the studied method is also includes in this paper. From the study done it is observed that depending upon the raw image and the application different features are exploited, hence implementing one of the techniques will not offer a high efficiency. This is mainly due to the fact that naturally captured images have varying features.

Thus a hybrid approach is required technique that can efficiently identify the text regions using varied text features.


Lungu

Louis Boisgibault
email: louis.boisgibault@paris-sorbonne.fr

How could the EU benefit from the COP 21 agreement?

Keywords: RCEPB, COP21, renewable energy, energy efficiency

The EU, one of the 196 Parties of Paris agreement was at the forefront of efforts towards the climate deal. W ill it be able , 1-to update on time the existing directives EU ETS (1), EPBD (2), RES (3), EED(4); 2-to transpose correctly and on time the COP21 agreement into the 28 Member States' national laws and last but not last , 3-to benefit from COP21 ?

In December 12 th 2015, 196 parties (195 countries and the European Union) adopted a universal, legally binding global climate deal at the 21 st Conference of Parties (COP21) in Paris. The Paris agreement sets out a global action plan to put the world on track to avoid dangerous climate change by limiting global warming to well below 2°C.

The agreement was deposited at the United Nations in New York City and has been opened for signature for one year since 22nd April 2016. The European Union signed it that first day with 174 countries. The EU is at the forefront of international efforts towards the global climate deal with, for instance, its new Climate & Energy Package in October 2014. But how will the transposition be done? The agreement will enter into force in 2020 after 55 countries that account for at least 55% of global emissions have deposited their instruments of ratification. Will the Paris Agreement be completely and accurately transposed into EU Member States' national laws and when? Experience shows that EU Member States' enforcement performances have been questionable in the past with transposition deficits, compliance deficits and delays. For complex topics such as Emission Trading Scheme, Renewable Energy and Energy Efficiency, where a lot has already been done, COP21 brings a supplementary level of authority forcing existing directives EU ETS (1), EPBD (2), RES (3), EED(4) to comply and to readjust their objectives for 2030 with the best path for 2050.

The benefit of all these efforts is a key question not only for the European Union but for its citizens. W ith a growing distrust in European institutions , disagreements between Member States, the possible Brexit and a lack of financing, the question of good implementation and enforcement is asked. How to implement a world agreement in European territories, cities, towns and rurality, so people, businesses and local authorities can see positive changes. How to reconcile this top-down approach where territories must comply with the need to promote local initiatives and people's involvement? This suggests that a very complex process is underway and that the European Union under carbon constraint (5) will face many difficulties to fully implement the Paris Agreement but there is no other choice. It is probably too early to assess all the benefits of an ambitious agreement that has not yet entered into force. Prospective simulations suggest that a new era is to come. For the construction industry, the benefits are for those who will scale up low-carbon developments, adopt novel building bio-based materials, combine new information and communication technologies with building integrated energy generations and offer solutions for sober energy consumptions.


Thierry Barbot
email: thierry.barbot@univ-avignon.fr

Carlos Maquera
email: cmaquera@icmc.usp.br

NIL-ANOSOV ACTIONS

Keywords: . 2000 Mathematics Subject Classification. Primary: 37C85 Anosov action, compact orbit, closing lemma, transitivity, Anosov flow

Introduction

A locally free action φ of a group G on a closed manifold M is said to be Anosov if there exists α in G, the Lie algebra of G, such that g := φ(exp α, •) is normally hyperbolically with respect to the orbit foliation. In our serie of papers, we focus on the case of codimension one Anosov actions, i.e. Anosov actions admitting a normally hyperbolic element for which the unstable direction has dimension one. This notion is very classical when the group G is R, i.e. in the case of Anosov flows. Until recently, there have been many works establishing what can be the behavior of a codimension one Anosov flow in a given manifold, even aiming to describe the Anosov flow when the ambient manifold is prescribed, with emphasis in dimension 3 (see [START_REF] Asaoka | On Invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture[END_REF][START_REF] Barbot | Caractérisation des flots d'Anosov en dimension 3 par leurs feuilletages faibles[END_REF][START_REF] Barbot | Flots d'Anosov sur les variétés graphées au sens de Waldhausen[END_REF][START_REF] Barbot | Mise en position optimale d'un tore par rapport à un flot d'Anosov[END_REF][START_REF] Barbot | Pseudo-Anosov flows in toroidal manifolds[END_REF][START_REF] Barbot | Classification and rigidity of totally periodic pseudo-Anosov flows in graph manifolds[END_REF][START_REF] Béguin | Building Anosov flows on 3-manifolds[END_REF][START_REF] Béguin | A spectral-like decomposition for transitive Anosov flows in dimension three[END_REF][START_REF] Bonatti | Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension[END_REF][START_REF] Fenley | Anosov flows in 3-manifolds[END_REF][START_REF] Fenley | The structure of branching in Anosov flows of 3-manifolds[END_REF][START_REF] Franks | Anosov diffeomorphisms. Global Analysis[END_REF][START_REF] Franks | Anomalous Anosov flows[END_REF][START_REF] Ghys | Codimension one Anosov flows and suspensions[END_REF][START_REF] Ghys | Rigidité différentiable des groupes fuchsiens[END_REF][START_REF] Ghys | Déformations de flots d'Anosov et de groupes fuchsiens[END_REF][START_REF] Matsumoto | Codimension one Anosov flows[END_REF][START_REF] Simic | Codimension one Anosov flows and a conjecture of Verjovsky[END_REF][START_REF] Verjovsky | Codimension one Anosov flows[END_REF]).

In our first paper [START_REF] Barbot | Transitivity of codimension one Anosov actions of R k on closed manifolds[END_REF], we intend to extend this analysis in the case where G is still abelian, but of bigger dimension. We prove in particular that if the dimension of the ambient manifold exceeds by at least 3 the dimension of the acting abelian group, then the action is topologically transitive. We also proposed the following conjecture, natural extension of a previous well-known conjecture: Generalized Verjovsky Conjecture. Let G be an abelian Lie group of dimension k. Every codimension one faithful Anosov action of G on a manifold of dimension ≥ k + 3 is topologically conjugate the suspension of an Anosov action of Z k on a closed manifold.

Meanwhile, V. Arakawa completely elucidated in his Ph. D Thesis the case of Anosov actions of R k on a closed manifold of dimension k + 2 : such an action reduces through a flat bundle to an Anosov flow on a 3-manifold ( [START_REF] Arakawa | Sobre classificação de ações Anosov de R k em (k + 2)-variedades fechadas[END_REF]).

Afterwards, in our work [START_REF] Barbot | Algebraic Anosov actions of nilpotent Lie groups[END_REF] aiming to generalize Tomter's classification of algebraic Anosov flows ( [START_REF] Tomter | Anosov flows on infra-homogeneous spaces[END_REF][START_REF] Tomter | On the classification of Anosov flows[END_REF] in the context of actions of abelian groups of higher dimension, it became clear that the natural setting was the case where the group G is nilpotent. In particular, the Generalized Verjovsky Conjecture above can be replaced by the similar statement where the group G is nilpotent, not necessarily abelian. Actually, it follows from the present paper that a nilpotent Lie group admitting a faithful Anosov action of codimension one on a closed manifold is necessarily abelian (Theorem 5).

We therefore started the analysis of Anosov actions of non-abelian Lie groups, which is the topic of the present paper. The key point one needs in order to undertake such an analysis is that the Anosov action is good in the sense of [START_REF] Tavares | Anosov actions of nilpotent Lie groups[END_REF], i.e. that the Anosov splitting associated to a partially hyperbolic element should be preserved by the other elements of G. It is for this reason that previously most authors only considered the case where the partially hyperbolic element lies in the center of G, for which this condition is obviously satisfied. Here, we observe that this property still hold if G is only nilpotent 1 . Actually, we point out a more general phenomenon: in order to have the G-invariance of the Anosov splitting, we only need the fact that the partially hyperbolic element belongs to the nilradical of G : the Anosov action is then called Nil-Anosov (see Theorem 1).

However, the study of Nil-Anosov actions is limited by the fact that the density of periodic orbits in the nonwandering set may fail in this more general situation. It is for this reason that we quickly restrict ourselves to the case of nilpotent Lie groups. Nevertheless, we consider that the study of Nil-Anosov actions is interesting in itself.

In section 3, we prove a spectral decomposition Theorem for Anosov actions of nilpotent Lie groups: the nonwandering set of such an action can be decomposed in a finite number of basic sets, each of them being topologically transitive (Theorem 4). Observe

1 In [START_REF]Hirsch Foliations and noncompact transformation groups[END_REF], M.W. Hirsch already made this observation, but never afterwards published a paper including the proofs of the results stated in [START_REF]Hirsch Foliations and noncompact transformation groups[END_REF].

that such a result is not merely an adaptation of the arguments involved in the case of diffeomorphisms or flows: the notion of nonwandering set is itself delicate and our analysis involves for example the fact that the nonwandering set of any element of G coincide with the nonwandering set of the entire group G (Remark 7).

One of the most important result in section 3 is the non-trivial fact when the Anosov action is faithful, then the holonomy representations of each leaf of the weak foliations are faithful (Proposition 6).

This fact is the principal element of the proof of Theorem 5 mentioned above: the study of codimension one actions of nilpotent Lie groups reduces to the case of abelian Lie groups. Therefore, the topological transitivity of these actions in higher dimensions follows from [START_REF] Barbot | Transitivity of codimension one Anosov actions of R k on closed manifolds[END_REF], but here we propose an alternative simpler proof of this fact (see Theorem 6). Finally, we conclude with an alternative and much simpler proof of Arakawa's Theorem: Anosov actions of a nilpotent Lie group of dimension k on a manifold of dimension k + 2 is a Nil-extension of an Anosov flow on a 3-manifold.

Acknowledgments. This paper was partly written while the second author stayed at Laboratoire de de Mathématiques, Université d'Avignon et des Pays de Vaucluse. It has been concluded while the second author was visiting the University of Campinas (UNI-CAMP) and supported by the french-brazilian program "Chaires franco-brésiliennes dans l'état de São Paulo". Pr. Régis Varão contributed to the conclusion of the paper, mainly in the elaboration of Theorem 7.

Preliminaries

Definitions and notations.

Let G be a connected, simply connected Lie group of dimension k, let G be the Lie algebra of G, and let M be a C ∞ manifold of dimension n + k, endowed with a Riemannian metric • , and let φ be a locally free smooth action of the simply connected Lie group G on M .

We will use the following usual simplified notations, denoting φ(g, x) by g.x or gx for every x in M and every g in G. We will however denote by φ g the associated diffeomorphism of M , when non evaluated on an element of M, in order to make a distinction with the element g of G. We denote by G.x or O G (x) := {gx, g ∈ G} the orbit of x ∈ M : in the second notation, we keep in mind that the orbit is the leaf of a foliation, the orbit foliation that we denote by O G . We will also use the simplified notation O and O(x) when there is no ambiguity on the group G. We denote by ∆ x := {g ∈ G : gx = x} the isotropy group of x. The action can be thought as a morphism G → Diff(M ), the kernel of which is called the kernel of φ: the kernel is the intersection of all the isotropy groups. The action is said to be locally free if the isotropy group of every point is discrete.

We can also see this data as an injective Lie morphism from G into the Lie algebra of vector fields of M. For every g in G, the flow (t, x) → exp(tg)x will be called the flow generated by g, denoted by φ g , and the vector field inducing this flow is denoted by X g .

Let F be a continuous foliation on a manifold M . We denote the leaf that contains p ∈ M by F(p). For an open subset U of M , let F| U be the foliation on U such that

(F| U )(p) is the connected component of F(p)∩U containing p ∈ M . A coordinate function ϕ = (x 1 , • • • , x n ) on U is called a foliation coordinate of F if x m+1 , • • • , x n are constant functions on each leaf of F| U ,
where m is the dimension of F. A foliation is of class C r if it is covered by C r foliation coordinates. We denote the tangent bundle of M by T M . If F is a C 1 foliation, then we denote the tangent bundle of F by T F.

We fix once for all the Riemannian metric • , and denote by d the associated distance map in M . We also fix a metric | • | on the Lie algebra G.

Anosov actions.

Let T O the k-dimensional subbundle of T M that is tangent to the orbits of φ. Definition 1. .

(1) A G-splitting of T M is a pair ξ = (E 1 , E 2 ) of subbundles of T M such that:

T M = E 1 ⊕ T O ⊕ E 2 (2) Two G-splittings ξ = (E 1 , E 2 ) and ξ = (E 1 , E 2 ) are transverse one to the other if (E 1 , E 2 ) and (E 1 , E 2 ) are also G-splittings. Clearly, if ξ = (E 1 , E 2 ) and ξ = (E 1 , E 2 ) are transverse, then dimE 1 = dimE 1 and dimE 2 = dim E 2 .
Observe that we don't require the splitting to be G-invariant.

Definition 2. .

(1) We say that a ∈ G is an Anosov element for φ if g = φ exp a acts normally hyperbolically with respect to the orbit foliation. That is, there exists real numbers λ > 0, C > 0 and a continuous Dg-invariant G-splitting ξ = (E ss , E uu ) such that if, m denotes the co-norm operator, then:

Dg n | E ss , Dg -n | E uu , Dg n | E ss m(Dg n | T O ) , Dg n | T O m(Dg n | E uu )
≤ Ce -λn , for all n ≥ 0.

In the terminology of [START_REF] Hirsch | Invariant manifolds[END_REF], it means that g is 1-normally hyperbolic to O. In this case, we say that the element a of G and the element exp a of G are ξ-Anosov. The splitting ξ is called a g-hyperbolic splitting. Observe that given an Anosov element g, the g-hyperbolic splitting is unique. (2) Call φ an Anosov action if some a ∈ G is an Anosov element for φ.

(3) The action φ is a codimension-one Anosov action if E uu is one-dimensional for some Anosov element a ∈ G.

Note that the splitting ξ depends on the Anosov element a. By Hirsch-Pugh-Shub theory in [START_REF] Hirsch | Invariant manifolds[END_REF] we obtain that the ξ is Hölder continuous and the subbundles

E ss , E uu , T O ⊕ E ss , T O ⊕ E uu are integrable. The corresponding foliations, F ss [ξ], F uu [ξ], F s [ξ], F u [ξ]
, are called the strong stable foliation, the strong unstable foliation, the weak stable foliation, and the weak unstable foliation, respectively. The orbits of φ are the leaves of a central foliation O that we call orbit foliation.

We insist on the fact that these foliations are not necessarily G-invariant: consider for example the case the affine group acting on the left on the quotient SL(2, R)/Γ where Γ is a cocompact lattice of SL(2, R) and G the group Aff of upper triangular matrices. In this case, any non-trivial diagonal matrix is an Anosov element g of Aff for which, let's say, the unstable bundle E uu is trivial. Then the stable bundle E ss is g-invariant, but not Aff-invariant. The C-orbit of a point x, denoted by C x , is the set comprising points of the form exp(g 1 )... exp(g N )x where the g i 's are elements of C in arbitrary numbers.

We consider Anosov subcones as approximations of one parameter subgroups: the Corbit of a point x is contained in the orbit O x and is made of the points of the form c(1) where c : [0, 1] → M is a differentiable curve such that c(0) = x and at any time t, the derivative ċ(t) is tangent to the orbit of the flow φ g where g is an element of C. The thiner the cone is, the closer are the C-orbits to the trajectory of φ g for any element g of G.

The following remark is a consequence of the structural stability of Anosov elements (see Theorems (7.1), (7.3) and the Remark after Theorem (7.3) in [START_REF] Hirsch | Invariant manifolds[END_REF]). Remark 2. The set A = A(φ) of Anosov elements for φ is an open subset of G. More precisely, the map associating to an element a in A its Anosov splitting ξ(a) is continuous.

Remark 3. On the other hand, there is a neighborhood U of ξ(a) in the space of splittings (with prescribed stable/unstable dimensions) such that ξ(a) is the only a-invariant splitting in U .

More precisely: let ξ = (E 1 , E 2 ) be any G-splitting transverse to ξ = (E ss , E uu ). Then, at every point x of M , the subspace E 1

x is the graph of a linear endomorphism E ss x → E uu

x ⊕T O x , hence ξ provides a section σ of Hom(E ss , E uu ⊕T O). If ξ is exp(a)-invariant, then σ is exp(a)-equivariant. But it follows easily from the Anosov property that the unique exp(a)-equivariant section of Hom(E ss , E uu ⊕ T O) is the null one. Therefore we have E 1 = E ss , and, similarly, E 2 = E uu .

Nil-Anosov actions.

Assume that G has a nontrivial nilradical N. We call φ a Nil-Anosov action if some a ∈ N , the Lie algebra of N, is an Anosov element for φ. In this case a is call a Nil-Anosov element for φ.

Theorem 1. If φ is a Nil-Anosov action, then for any Anosov element a of the nilradical N , the hyperbolic splitting ξ for which a is Anosov is preserved by the action of G. In particular, every foliation

F ss [ξ], F uu [ξ], F s [ξ], F u [ξ] is G-invariant.
The proof of this Theorem is based on this fundamental lemma, that we will use several time in this paper: Lemma 1. (Ascending chains of normalizers stabilize) Let H 0 be a Lie subalgebra in a nilpotent Lie algebra N . Let (H i ) (i∈N) be the infinite sequence obtained by taking the successive normalizers: for every integer i, H i+1 is the normalizer of H i in N . Then, this sequence eventually stabilizes to N : there is an integer k such that H k = N .

Proof. Nilpotent groups satisfies the normalizer condition: for every Lie subalgebra H, the normalizer N (H) contains strictly H, unless H is already the entire Lie algebra N . It follows that the dimension of H i increases with i, until the equality H k = N holds. Lemma 2. Let H be a Lie subalgebra of G containing a ξ-Anosov element and assume that for every a in H the splitting ξ is exp(a)-invariant. Then, for every h in the normalizer

N G (H) in G the splitting ξ is exp(h) invariant.
Proof. Let a = exp(a) be an Anosov element in H = exp(H). Then, for all h in N G (H) the conjugate exp(h)a exp(-h) is exp(h)(ξ)-Anosov. If h is small, the splittings ξ and exp(h)(ξ) are close one to the other, in particular, transverse one to the other. Since by hypothesis ξ is exp(h)a exp(-h)-invariant (since exp(h)a exp(-h) lies in H), it follows from Remark 3 that we have:

exp(h)(ξ) = ξ,
The lemma follows since every element of exp(N G (H)) is an iterate of some exp(h) with h small.

Proof of Theorem 1. Let H 0 be the subalgebra of N generated by the Nil-Anosov element a. We define, inductively, H i+1 = N N (H i ) for all i in N. By Lemma 2 we have that ξ is preserved by H i for every i in N. On the other hand, by Lemma 1, there exists k in N such that N = H k . Finally, by applying Lemma 2 to N G (N ) and using the fact that N G (N ) = G, because N is the nilradical of G, we obtain that G preserves ξ.

Proposition 1. Let a 0 be an element of N which is ξ-Anosov. Then, the set A ξ of ξ-Anosov elements of G (not necessarily in the nilradical) is an open convex cone, which is a connected component of the set A of Anosov elements of G.

Proof. By Remark 1, A ξ is a convex cone. According to Remark 2, A is an open neighborhood of A ξ . The map associating to an element a in A its Anosov splitting ξ(a) is continuous; and it follows from Theorem 1 and Remark 3 that it is locally constant at every point in A ξ . It follows that A ξ is open in A. The proposition follows since A ξ is clearly closed in A.

Connected components of A are called chambers. From now we will assume that every Nil-Anosov action comes with a specified Nil-Anosov element a 0 . We denote by ξ the splitting for which a 0 is ξ-Anosov, and by A 0 = A ξ the chamber containing the preferred Anosov element a 0 . The splitting ξ defines the stable/unstable foliations common to every element in A 0 , that we will denote by F s , F u , F ss and F uu , dropping the notation [ξ]. According to Theorem 1, all these foliations are G-invariant.

For all δ > 0, F i δ (x) denotes the open ball in F i (x) centered at x with radius δ with respect to the restriction of • to F i (x), where i = ss, uu, s, u.

Theorem 2 (local product structure). Let φ : G × M → M be a Nil-Anosov action. There exists a δ 0 > 0 such that for all δ ∈ (0, δ 0 ) and for all x ∈ M, the applications

[•, •] u : F s δ (x) × F uu δ (x) → M ; [y, z] u = F s 2δ (z) ∩ F uu 2δ (y) [•, •] s : F ss δ (x) × F u δ (x) → M ; [y, z] s = F ss 2δ (z) ∩ F u 2δ (y)
are homeomorphisms onto their images.

Remark 4.

(1) Every leaf of F ss or F uu is a plane, i.e. diffeomorphic to R for some . This is a straightforward observation, since every compact domain in a leaf of F ss (respectively of F uu ) shrinks to a point under positive (respectively negative) iteration by φ exp a 0 . (2) Let F be a weak leaf, let us say a weak stable leaf. For every strong stable leaf L in F , let Γ L be the subgroup of G comprising elements a such that φ a (L) = L, and let ω L be the saturation of L under φ. Thanks to Theorem 2 we have:

• ω L is open in F , • Γ L is discrete. Since F is connected, the first item implies F = ω L : the φ-saturation of a strong
leaf is an entire weak leaf. The second item implies that the quotient P = G/Γ L is a manifold. For every x in F , define p F (x) as the equivalence class aΓ L such that x belongs to φ a (L). The map p F : F → P is a locally trivial fibration and the restriction of p F to any φ-orbit in F is a covering map. Since the fibers are contractible (they are leaves of F ss , hence planes), the fundamental group of F is the fundamental group of P , i.e. Γ L for any strong stable leaf L inside F .

Observe that if F ss is oriented, then the fibration p F is trivial: in particular, F is diffeomorphic to P × R p , where p is the dimension of F ss . Of course, analogous statements hold for the strong and weak unstable leaves.

(3) In the continuation of the previous items, let F be a weak stable leaf, x a point in F and L the strong stable leaf through x. Since p F : F → P is a locally trivial fibration and the fibers L are contractible, loops in F based at x are all homotopic to a loop ω h x obtained by composing a path from [0, 1] into O x of the form t → exp(th) where h ∈ G satisfies (exp h)L = L with a path joining exp(h).x to x. The holonomy of such a loop does not depend on the choice of the final path joining exp(h).x to x since L is contractible; this holonomy defines a germ at x of homeomorphism of F uu (x) which only depends on x and h. We denote it by h a x .

Remark 5. We will also need a local description of fixed points of Anosov elements: let a = exp(a) be an ξ-Anosov element of an Anosov action of G. Let x be a fixed point of φ a . Let D be a small disk containing x transverse to the orbit foliation O. There is a small neighborhood U of x in M and a neighborhood V of e (the identity element) in G such that, for every z in U , there is one and only one element b in V such that bz belongs to D. Moreover, reducing U if necessary, we can assume that for every b in V the product ba is Anosov. Let D ⊂ D be a subdisk such that φ a (D ) is contained in U . Then there is a "first return map" ϕ : D → D and a "variation of time of first return map" α : D → V uniquely defined by: ϕ(z) = α(z)a.z ∈ D. Observe that the map ϕ could also be defined as the holonomy of O along the periodic orbit of the flow φ a containing x.

The weak foliations F s and F u are transverse to D and thus define two foliations G s and G u , transverse to one another, of supplementary dimensions, and preserved by the holonomy map ϕ. Let s 0 (respectively u 0 ) the leaf of G s (resp. G u ) through x. They are both ϕ-invariant, the action of ϕ on s 0 is contracting, and the action on u 0 is expanding. By considering the map from D into s 0 × u 0 associating to a point y the pair (p, q), where p is the intersection between G u (y) and s 0 , and q the intersection between G s (y) and u 0 (cf. Theorem 2), we get that x is a saddle fixed point of ϕ. In particular, x is an isolated fixed point of ϕ. Remark 6. If H is a Lie subgroup of N containing a Nil-Anosov element, then every uniform lattice in H contains Anosov elements. In particular, the isotropy subgroup of a point in M whose H-orbit is compact contains Anosov elements. Indeed, if ∆ is an uniform lattice of H, then, by a result in [3, Theorem (5.1)], ∆ contains a log-lattice, i.e. a finite index sub-lattice ∆ ⊂ ∆ such that log ∆ is a lattice of the vector space H. On the other hand, let h be a Nil-Anosov element in H and let A 0 be the chamber in G containing the Anosov element h. Then, since Proposition 1 implies that H ∩ A 0 is an open convex cone in H, the set (H ∩ A 0 ) ∩ log ∆ is not empty. Consequently, ∆ ⊂ ∆ contains Anosov elements, and thus, our assertion follows.

Examples.

In this subsection, we give examples of general Anosov actions, not necessarily Nil-Anosov. We start showing how to produce new Anosov actions from the data of Anosov actions.

Products of Anosov actions. If φ

1 : G 1 × M 1 → M 1 and φ 2 : G 2 × M 2 → M 2 are two Anosov actions, then the product action of G 1 × G 2 on M 1 × M 2 is clearly Anosov.
The nilradical of G 1 × G 2 is the product of the nilradicals of G 1 and G 2 , and an element (g 1 , g 2 ) of G 1 × G 2 is Anosov for φ 1 × φ 2 if and only if g 1 is Anosov for φ 1 and g 2 is Anosov for φ 2 . It follows that the product action is Nil-Anosov if and only if

φ 1 : G 1 × M 1 → M 1 and φ 2 : G 2 × M 2 → M 2 are both Nil-Anosov.
This remark includes the limit case where one of the action, let's say φ 2 : G 2 ×M 2 → M 2 , is transitive, i.e. an action of G 2 by left translations on a compact quotient G 2 /Λ 2 . This is partially generalized in the next section.

2.4.2. Nil-extensions. We consider any extension:

0 → H → G p → G → 0
where H is a nilpotent Lie group. A locally free action φ :

G × M → M is a Nil-extension of φ : G × M → M if there is a G-equivariant map π : M → M whose fibers are precisely the H-orbits: π(ĝ.x) = p(ĝ).π(x).
We furthermore require that the map x → H x where H x is the stabilizer of x for the action of H is continuous as map from M into the space of lattices of H.

In other words, π is a locally trivial fibration over M, with fibers homeomorphic to compact quotients of H. It induces an identification between the orbit space H\ M and M , and φ is the induced action.

A Nil-extension introduces no additional dynamical feature. In particular, elements of H have a "trivial" transversal action; they cannot be Anosov relatively to the action φ. Hence, clearly, φ is (Nil-)Anosov if and only if φ is (Nil-)Anosov.

A fundamental type of Nil-extensions are the central extensions, i.e. Nil-extensions for which N is in the center of G. Observe that central extensions are not necessarily product; see Starkov's example described in [11, Section 3.2.3].

2.4.3.

Affine group of the real line. Let Aff be the subgroup of SL(2, R) made of upper diagonal matrices: we have already considered previously the action of Aff on compact quotients SL(2, R)/Γ as an example of Anosov action for which the splitting of Anosov elements is not preserved by the entire group. This Anosov action is not Nil-Anosov. Observe that it admits no compact orbits.

2.4.4. Engel subalgebras. Let G be a real Lie group, not necessarily semisimple, and let G be its Lie algebra. For any element h of G we denote by ad h the adjoint action of h on G.

We denote by L 0 (h) the 0-characteristic subspace of ad h : it is a subalgebra of G, called a Engel subalgebra. We actually have a splitting of G by characteristic subspaces:

G = L 0 (h) ⊕ L s (h) ⊕ L u (h) ⊕ L i (h)
where L s (h) (resp. L u (h)) is the sum of characteristic subspaces associated to eigenvalues with negative real part (resp. positive real part), and where L i (h) is the sum of characteristic subspaces associated to purely imaginary eigenvalues.

The sum [START_REF] Barbot | Algebraic Anosov actions of nilpotent Lie groups[END_REF], the authors study in detail these examples when H is nilpotent.

H = L 0 (h) ⊕ L i (h)
2.5. Nil-Anosov actions are Nil-faithful up to Nil-extensions. We can reduce any action φ : G × M → M to a faithful one simply by replacing G by G/ ker φ and considering the induced action on M. On the other hand, many aspects of nilpotent Lie groups are simplified when they are simply connected: the exponential map is a diffeomorphism, and one can more easily manipulate subgroups of N when it is simply connected. By taking the universal covering if necessary, we can easily reduce to the case where G and its nilradical are simply connected, but then we may loose the faithfulness of the action.

Nevertheless, the following proposition shows that, up to Nil-extensions, we can keep the fact that the nilradical is simply connected. Proof. Let φ : G × M → M be a Nil-Anosov action which is not Nil-faithful. Let Λ be the intersection between the nilradical N and the kernel of φ. Then Λ is discrete in N and stable by conjugacy: it follows that Λ is in the centralizer Z(G) of G. Since N is nilpotent and simply connected, Z(N ) = Z(G) ∩ N is isomorphic to R m for some m; let H be the minimal connected Lie subgroup of Z(N ) containing Λ. Then the torus T := Λ\H acts freely and properly on M, and the orbit space M := T \M is equipped with an Anosov action of H\G whose nilradical H\N is simply connected. The quotient map M → M is a central extension.

The Lemma is obtained by repeating the argument until the action on the quotient space is Nil-faithful.

Anosov actions of nilpotent Lie groups

From now on, we restrict to the case where the Lie group G is nilpotent, i.e. equal to its nilradical N . In this section, we will observe that compact orbits play a central role in the dynamical behavior of Anosov actions of nilpotent Lie groups.

Convention: We denote by Comp(φ) the set of compact φ-orbits.

By Lemma 3, we can assume without loss of generality that N = G is simply connected (all our results are insensitive to Nil-extensions).

3.1. Compact orbits. Most of the content of this section and of the following section 3.2 is announced in [START_REF]Hirsch Foliations and noncompact transformation groups[END_REF], but the proofs have not been published anywhere.

Theorem 3. Let φ be an Anosov action of a nilpotent Lie group N . A φ-orbit is compact if and only if it contains a point fixed by an Anosov element.

The Theorem will be proved by induction involving the following Lemma. Lemma 4. Let H be a subgroup of N containing an Anosov element a = exp(a) and let x 0 be an element of M. Let N (H) denote the normalizer in N of H. If the H-orbit of x 0 is compact, then the N (H)-orbit of x 0 is compact.

Proof. Assume that A 0 is the chamber containing the Anosov element a in H. Let O H (x 0 ) be the H-orbit of x 0 , and let ∆ 0 be the isotropy subgroup of x 0 by the H-action, that is ∆ 0 is the uniform lattice in H which is given by ∆ 0 = {h ∈ H : φ h (x 0 ) = x 0 }. Let x ∞ be a point in the closure of the N (H)-orbit of x 0 , then there exists a sequence of elements z n of N (H) such that lim

n→∞ φ zn (x 0 ) = x ∞ .
As z n H = Hz n , then the H-orbit of φ zn (x 0 ) is exactly equal to φ zn (O H (x 0 )), in particular, it is compact. Hence, for every n the isotropy subgroup of φ zn (x 0 ) by the H-action, which we denote by ∆ n , is a uniform lattice in H. Moreover, for every n we have

∆ n = z n ∆ 0 z -1 n .
Since N is nilpotent, the adjoint action of z n on H is unimodular. In particular, all the lattices ∆ n have the same co-area in H. Moreover, since the action on the compact manifold M is locally free, there is a uniform bound from below for the size of elements in ∆ n . By Mahler's criterion for nilpotent Lie groups with a Q-structure (which is the case for H since it contains a cocompact lattice) (see [START_REF] Moskowitz | An extension of Mahler's theorem to simply connected nilpotent groups[END_REF]), up to a subsequence, we can suppose that the sequence ∆ n converges to some uniform lattice ∆ ∞ in H. In particular, for every h ∞ in ∆ ∞ , there is a sequence of elements h n of ∆ n converging in H to h ∞ . Let γ n be the element in ∆ 0 such that h n = z n γ n z -1 n . Then, by continuity of the map φ :

N × M → M : h ∞ .x ∞ = lim n→∞ h n .(z n .x 0 ) = lim n→∞ z n γ n z -1 n z n .x 0 = lim n→∞ z n γ n .x 0 = lim n→∞ z n .x 0 = x ∞ .
This shows that ∆ ∞ lies in the isotropy subgroup of x ∞ by the H-action and, consequently, the H-orbit of x ∞ is compact.

We claim that x ∞ is in the φ-orbit of x 0 . In fact, by Remark 6, there is an Anosov element a in H such that: φ a (x ∞ ) = x ∞ (in other words, a is in ∆ ∞ ) and log a is an element of the chamber A 0 . As defined in Remark 5, we consider the local return map ϕ : D → D along the φ a -orbit of x ∞ , where D and D are small disks transverse to the φ-orbit of x ∞ , and x ∞ is the unique fixed point for ϕ, of hyperbolic type.

Let V H be a small neighborhood of the identity in H such that the map h → hx from V H to M is injective for every element x of D. When x describes D , the pieces of orbits V H .x are disjoint one from the other, and have an area uniformly bounded from below. Since the orbits O H (z n .x 0 ) have all the same area, it follows that the number of intersections between D and every O H (z n .x 0 ) is bounded from above, uniformly in n, by some integer m.

On the other hand, the points φ zn (x 0 ) accumulate on x ∞ , hence belong to D for n sufficiently big. Then, if n is sufficiently big, the iterates ϕ k (φ zn (x 0 )) belongs to D for all intergers k between 0 and m + 1. By definition of m, two of these iterates must be equal, but since x ∞ is a fixed point of ϕ of hyperbolic type, it is possible only if φ zn (x 0 ) = x ∞ . Hence, x ∞ belongs to the φ-orbit of x 0 , and thus, our claim is verified.

Let g be an element of N such that x ∞ = g.x 0 . This implies that ∆ ∞ = g∆ 0 g -1 , hence g is an element of N (H) since ∆ ∞ and ∆ 0 are both lattices in H.

Thus, we have proved that every point x ∞ in the closure of the N (H)-orbit of x 0 is in the N (H)-orbit of x 0 : this orbit is compact.

Proof of Theorem 3. One of the implication is a direct consequence of Remark 6. Let us prove the other implication: let x 0 be an element of M fixed by an Anosov element a in N. We will show that the φ-orbit of x 0 is compact. In fact, let a be the element in N such that exp a = a. We consider H 0 , the subgroup of N which is the one-parameter group associated to a. It is clear that the H 0 -orbit of x 0 is compact (a circle). We define, inductively, H i+1 = N (H i ) for all i in N. By Lemma 4 we have that the H i -orbit of x 0 is compact for every i in N. Hence, by Lemma 1, there exists k in N such that N = H k and the Theorem follows.

Let us fix a Haar measure on N , so that one can estimate the area of every φ-orbit. Proposition 2. For every C > 0, there is only a finite number of compact φ-orbits of area ≤ C.

Proof. Assume by contradiction the existence of an infinite sequence of distinct compact orbits O n of area ≤ C. For each of them, let ∆ n be the isotropy group of O n : it is a lattice in N .

Since φ is locally free, the length of elements of ∆ n is uniformly bounded from below, independantly from n. Once more, by the Mahler's criterion extended to nilpotent Lie groups ( [START_REF] Moskowitz | An extension of Mahler's theorem to simply connected nilpotent groups[END_REF]), it ensures that, up to a subsequence, the ∆ n converges to some lattice ∆ ∞ . In particular, for every a ∞ in ∆ ∞ , there is a sequence of elements a n of ∆ n converging in N to a ∞ . Furthermore, according to Remark 6, we can select a ∞ to be Anosov. Up to a subsequence, we can also pick up a sequence of elements x n in each O n converging to some x ∞ in M . Then, since φ an (x n ) = x n , at the limit we have φ a∞ (x ∞ ) = x ∞ . Since a ∞ is Anosov, the φ-orbit O ∞ of x ∞ is compact. Consider a local section Σ to φ containing x ∞ : the first return map on Σ along the orbit of φ a∞ is hyperbolic, admitting x ∞ as an isolated fixed point. On the other hand, by pushing slightly along φ, we can assume without loss of generality that every x n belongs to Σ. Since the a n converges to a ∞ , the φ an -orbit of x n approximates the φ a∞ -orbit of x ∞ , showing that the x n are also fixed points of the first return map. It is a contradiction, since they accumulate to the isolated fixed point x ∞ .

Spectral decomposition.

Definition 5 (Nonwandering set of an Anosov element). A point x ∈ M is nonwandering with respect to an Anosov element a if for any open set U containing x there is a real number t > 1, such that φ exp(ta) (U ) ∩ U = ∅. The set of all nonwandering points, with respect to a, is denoted by Ω(a). Let C be a strict Anosov subcone containing a. Observe that Ω(a) and Ω(C) are closed, and the inclusion Ω(a) ⊂ Ω(C) is obvious. If the φ-orbit of a point x is compact, then the restriction of the action of a on this orbit is a left translation on a compact right quotient of N by a lattice; hence this action preserves a volume form on the orbit and every point is nonwandering. The inclusions Comp(Φ) ⊂ Ω(a) ⊂ Ω(C) follow. Inversely: Proposition 3. For any strict Anosov subcone C, the union Comp(φ) of compact orbits of N is dense in Ω(C).

Proof. Let x ∈ Ω(C). We denote by C ≥1 the subset of C comprising elements with |-norm ≥ 1. For every ε > 0 let D ε be a small disc of size ε centered at x and transverse to the orbit foliation O, so that the restrictions G s and G u of the weak foliations to D are product; any leaf of G s intersects every leaf of G u in one and only one point -the existence of such a disc is a corollary of Theorem 2. We can assume that these discs are nested, i.e. that D ε ⊂ D ε if ε ≤ ε . Let U be a relatively compact symetric neighborhood of the identity in G such that the application U × D ε → M mapping (g, y) to φ g (y) is a diffeomorphism onto its image.

Since C is an open convex cone, reducing U if necessary, one can assume that for any a ∈ C ≥1 and any g in the closure of U, the product g exp(a) is Anosov. Moreover, there is a uniform bound C > 0 such that the restriction of dφ g exp(a) to E ss (respectively to E uu ) is Ce -λ|a| -contracting (respectively C -1 exp(λ|a|)-expanding).

By hypothesis, for any n, there is an element x n in D 1/n and an element a n in C such that φ an (x n ) lies in the neighborhood described above, i.e. has the form φ gn (y n ) where y n lies in D 1/n and g n in U .

We consider first the case where the real numbers |a n | are bounded from above, i.e. that, up to a subsequence, a n converge to some element a of C ≥1 . Since C is strict, a is Anosov.

Up to a subsequence, x is a fixed point of g -1 ∞ exp(a) where g ∞ is a limit of the g n 's. Hence, x fixed by an Anosov element, and thus contained in Comp(φ). The Lemma is proved in this case.

We are thus reduced to the other case, i.e. the case where, up to a subsequence, the real numbers |a n | converge to +∞. Consider the local unstable leaf G s (x n ): its image by the first return map from a neighborhood of x n in D 1/n into a neighborhood of y n in D 1/n is contracted, and therefore this first return map is well-defined. The composition of this map with the holonomy along G u defines a contracting map from G s (x n ) into itself, which therefore admits a fixed point. It means that there is an element h n of U such that h n exp(a n ) maps a point z n of D 1/n to a point p n in its unstable leaf G u (z n ). Now apply the same argument now to the reversed subcone -C, and replacing x n by p n : since the stable/unstable foliations are switched when we replace C by its opposite, we obtain that the first return map preserves a local stable leaf.

The intersection of theses local stable and unstable leaves is now a point fixed by an element of the form g exp(a n ) with g in Ū , hence Anosov. Therefore D 1/n contains a point fixed by an Anosov element, hence lying in Comp(φ). The Proposition follows since n is arbitrary.

Remark 7. It follows by proposition 3 that we have:

Ω(a) = Ω(C) = Comp(φ)
In particular, the nonwandering sets Ω(a) and Ω(C) are independent from the choice of the Anosov element a, the Anosov subcone C, and even of the preferred splitting ξ and preferred Anosov chamber A 0 . Hence we can denote the nonwandering set simply by Ω(φ) or Ω.

We will need the following refinement of Theorem 3. Recall Definition 3: Lemma 5. For any Anosov subcone C, and any compact orbit O x , there is an element of C fixing an element in O x .

Proof. It follows immediatly by repeating the argument used in the proof of Remark 6, replacing A 0 by C. Proposition 4. Let x, y be two elements of Ω. Assume that the intersections F u (x)∩F s (y) and F s (x) ∩ F u (y) are non-empty. Then, these intersections are both contained in Ω.

Proof. The argument is classical, but the proof more oftenly involves the notion of pseudoorbits, that we prefer to avoid here (see for exemple [START_REF] Shub | Global Stability of Dynamical Systems[END_REF]Proposition 8.11]). Since Comp(φ) is dense in Ω, and by continuity of the maps (x, y) → F u (x) ∩ F s (y) (cf. Theorem 2) on can assume that x and y are in Comp(φ).

Let C be a strict Anosov subcone in our preferred Anosov chamber A 0 (hence elements of C preserve the foliations F ss and F uu ). By Lemma 4, one can assume that x is fixed by some element a of C, and y by some element b of C.

Let p be a point in F uu (x) ∩ F s (y) and let U be a small product neighborhood of p. We aim to show that some non trivial element of C maps a point in U into U.

For k big enough, D = φ -ka (D) contains the point p. Reducing U if necessary, one can assume that U is the saturation under some arbitrarly small neighborhood V of e in N of an open neighborhood U of p in D (see Figure 1).

Let now q be a point in F ss (x) ∩ F u (y). Similarly, some iterate D = φ a (D) with k > 0 contains q. Let W be a neighborhood of q as depicted in Figure 1. In this figure, we draw two hatched regions W and W which have the following properties:

-

W ⊂ U , -W ⊂ W, -W = φ (k+ )a (W ), -W is saturated by the restriction G s U of F s to U , -W is saturated by the restriction G u W of F u to W.
We furthermore select W so that it is a product neighborhood, i.e. such that every leaf of G s W intersects every leaf of G u W . We now replace y by some N -iterate of itself so that q lies in F ss (y). Then we have p = g.p , where p is a point in F ss (y) ∩ F u (x) and g some element of N. By repeating the argument above, we get a region Z in W , saturated by the foliation G s W , which is mapped by a positive iterate φ pb to a region Z of g -1 (U ).

The closer U is, the bigger are the integers k, and p. Hence one can assume that g. exp(pb) exp((k + )a) is the image under exp of some element c of C ≥1 . Since W is G u Wsaturated, Z is G s W -saturated, and W a product neighborhood, the intersection W ∩ Z is non empty. It follows that any point in W ∩ φ -(k+ )a (W ∩ Z ) is mapped by φ c to a pont in gZ ⊂ U ⊂ U (see Figure 2). The Proposition is proved. Theorem 4 (Spectral decomposition). Let M be a closed smooth manifold and let φ be an Anosov action on M . The nonwandering set of φ can be partitioned into a finite number of φ-invariant closed subsets, called basic blocks:

Ω = i=1 Λ i
such that for every Anosov subcone C, every Λ i is C-transitive, i.e. for every open subsets U , V in M intersecting Λ i , there is an element x of U ∩ Λ i whose C-orbit in the meaning of Definition 3 meets V ∩ Λ i . In particular, there is an element of Λ i whose C-orbit is dense in Λ i .

Proof. Let Comp(φ) be the set of compact orbits of φ. By Proposition 3 we have Comp(φ) = Ω(φ). We define a relation on Comp(φ) by: x ∼ y if and only if F u (x) ∩ F s (y) = ∅ and F s (x) ∩ F u (y) = ∅ with both intersections transverse in at least one point. Note that this relation is preserved by φ. We want to show that this is an equivalence relation and obtain each Λ i as the closure of an equivalence class. Note that ∼ is trivially reflexive and symmetric. In order to check the transitivity suppose that x, y, z ∈ Comp(φ) and p ∈ F u (x) ∩ F s (y), q ∈ F u (y) ∩ F s (z) are transverse intersection points. There exists a ∈ A 0 such that ax = x, where a = exp(a). Since the iterations under φ a of a ball around p in F u (p) = F u (x) = φ a (F u (x)) accumulate on F u (y), we obtain that F u (x) and F s (z) have a transverse intersection. Analogously, we obtain that F s (x) and F u (z) have a transverse intersection.

By Theorem 2 any two sufficiently near points are equivalent, so by compactness we have finitely many equivalence classes whose (pairwise disjoint) closures we denote by Λ 1 , Λ 2 , . . . , Λ .

It remains to show that every Λ i is C-transitive for every Anosov subcone C ⊂ A 0 . It means that for any two open sets U and V in Λ i there exists a ∈ C such that φ exp(a) (U )∩V = ∅ -the existence of a dense C-orbit then follows by a classical argument (see for example [START_REF] Brin | Topological transitivity of one class of dynamical systems, and flows of frames on manifolds of negative curvature[END_REF]Lemma 1]).

The density of compact orbits in Λ i implies the existence of x ∈ U and a ∈ C such that φ a (x) = x (cf. Lemma 5).

Let y be an element in V ∩ Λ i ∩ Comp(φ). We have x ∼ y, hence the stable leaf F ss (y) intersects F u (x). Iterating by φ b where b is an Anosov element in A 0 fixing y, we observe as above that F u (x) contains points arbitrarly close to y; in particular, a point p in V . Now, according to Remark 3 item (2), there is an element g such that exp(g)p belongs to F uu (x). Then, the negative iterates p n := a -n exp(g)p converge to x.

Let n be big enough so that:

-na -g belongs to C, -p n := a -n exp(g)p belongs to U. By the Lie product formula, exp(-g/k+na/k) k for k going to +∞ converges to exp(-g)a n , hence maps the point p n of U to an element in V. Since every -g/k + na/k is an element of C, there is as required an element of C mapping an element of U to an element of V.

In order to conclude, we must show that this decomposition does not depend on our initial choice of preferred splitting ξ and the chamber A 0 . Let ξ be another splitting for which A ξ is non-empty. We obtain then a spectral decomposition Ω = i=1 Λ i such that for every Anosov subcone C ⊂ A ξ every Λ i is C -transitive. Let x and y be two elements of the same basic set Λ i , and let Λ j , Λ k be the basic sets for the splitting ξ containing respectively x and y. Then, there are sequences (x n ) n∈N and (y n ) n∈N in Ω, the first converging to x, the second to y, and such that y n lies in the C-orbit of x n . Then, since Λ i is isolated in Ω, every x n and y n lies in Λ i for n sufficiently big. But for the same reason, x n lies in Λ j and y n lies in Λ k for n sufficiently big. Since Λ j is G-invariant, we have Λ j = Λ k . Therefore, every ξ-basic set Λ i is contained in one ξ -basic set Λ j . The reverse inclusion is obtained by the same argument, exchanging the roles of ξ and ξ . The Theorem follows.

Definition 7. For every basic set Λ i we denote by F s (Λ i ) the union of the leaves F s (x) where x describes Λ i . Lemma 6. For every basic set Λ i and any Anosov element a of A 0 , an element x of M lies in F s (Λ i ) if and only if the distance between exp(ta)x and Λ i goes to zero when t → +∞.

Proof. If x lies in F s (Λ i ), since Λ i is φ-invariant, x lies in a strong stable leaf F ss (p) of some element in Λ i . Then the distance d(exp(ta)x, exp(ta)p) goes to 0 when t → +∞. It proves one implication of the statement.

Inversely, assume that d(Λ i , exp(ta)x) goes to zero. Assume by a way of contradiction that x does not belong to F s (Λ i ). Then, for t sufficiently big, x t := exp(ta)x lies in a neighborhood of Λ i where the local product property holds: there is a point p t in Λ i such that F uu δ (x t ) intersects F s δ (p t ). Since x / ∈ F s (Λ i ), we have x t / ∈ F s (p t ). Let r t be the distance in F uu (x t ) between x t and F s (Λ i ). At one hand we have the uniform bound r t ≤ δ for any t sufficiently big. On the other hand, since F s (Λ i ) is exp(a)-invariant, this distance increases exponentially with t. Contradiction.

Corollary 1. Every point in M lies in the stable leaf (respectively unstable leaf ) of a non-wandering point. In other words:

M = i=1 F s (Λ i ) = i=1 F u (Λ i )
Proof. Let a be an element of A 0 . Let V 1 , V 2 , ... , V be two by two disjoint open neighborhoods of the Λ i 's. The ω-limit set of the φ a -orbit of x is contained in Ω(a) and thus is covered by the disjoint union of the V i . Moreover, it is connected, hence it is contained in one and only one open domain V i , hence in V i ∩Ω(a) = Λ i . Therefore, the distance between exp(ta) and Λ i converges to 0. Then, it follows from Lemma 6 that x lies in F s (Λ i ).

The similar statement for the unstable foliation is proved by reversing the flow.

Remark 8. The decomposition Ω = i=1 Λ i is independant from the choice of the splitting ξ, but of course it is not the case for the decomposition

M = i=1 F s (Λ i ) = i=1 F u (Λ i ).
We will need later the follwing criteria:

Proposition 5. If every leaf of F s is dense in M , then Ω(φ) = M.
Proof. By Theorem 4, for any p in M there is x, y ∈ Ω(φ) such that p ∈ F u (x) ∩ F s (y). The hypothesis implies that the stable leaf F s (x) intersects F u (y). Hence by Proposition 4 we have p ∈ Ω(φ). Definition 8. Let G be the graph whose vertices are the basic sets Λ i , and such that there is an edge connecting Λ i and Λ j if and only if the closure of F u (Λ i ) contains Λ j (hence

F u (Λ j )), or the closure of F u (Λ j ) contains Λ i (hence F u (Λ i )).
Lemma 7. G is connected.

Proof. For every connected component C of C, let F (C) be the union of F u (Λ i ) where Λ i describes the set of basic sets in C. Let x be an element in the closure of F (C) : since the number of basic sets is finite, x is in the closure of F u (Λ i ) for some Λ i in C. Let Λ j be the basic set such that x ∈ F u (Λ j ) (cf. Corollary 1). Then F u (Λ j ) is contained in the closure of F u (Λ i ), hence Λ i is in C.

It follows that F (C) is closed. Let now C be any other connected component of G. Assume that F (C) ∩ F (C ) is non-empty: let x be an element in this intersection. Then x lies in some F u (Λ j ), and one proves as above that Λ j lies in C and in C . Hence C = C .

In other words, M is the disjoint union of the closed subsets F (C). The Lemma follows from this fact and the connectedness of M.

3.3.

Injectivity of the holonomy. Recall that in Remark 4 item (3) we have defined, for every point x and any element h of N satisfying (exp h)F ss (x) = F ss (x) a germ at x of homeomorphism h h

x of F uu (x), representing the holonomy transverse to the foliation F s of a certain loop ω h

x obtained by composing the orbit of x under φ h during the time [0, 1] and any path from exp(h)x to x in F ss (x). Moreover, every loop in the leaf F s (x) is homotopic to a loop ω h

x , hence every element of the holonomy group of the leaf F s (x) is represented by some h h

x . We call h h x the holonomy of h at x. Observe that if h belongs to A 0 , then h h x is expanding, in particular, non-trivial.

Definition 9. The Anosov action φ has injective stable holonomy if for every leaf F of F s the holonomy representation π 1 (F, x) → Homeo loc (F uu (x)) is injective. According to the discussion above, it is equivalent to the following requirement: for every non-trivial element h of G and every x ∈ M such that exp(h)F ss (x) = F ss (x) the holonomy h h x at x is non-trivial.

Similarly, one define injective unstable holonomy; the action has injective holonomy if it has stable and unstable injective holonomy.

Of course, Nil-extensions do not have injective holonomy: loops contained in the fibers are loops contained in orbits with trivial holonomy. The following proposition shows that it is the only obstruction to injective holonomy: Proposition 6. Faithful Anosov actions of nilpotent Lie groups have injective holonomy.

All this section is devoted to the proof of Proposition 6. More precisely, we prove that the action has injective stable holonomy, the injectivity of the unstable holonomy is then obtained by simply replacing the Anosov element by its opposite. Actually, we will prove the reverse statement, assuming that the action has no injective stable holonomy, and proving that then it is not faithful. Definition 10. .

(1) Let h be a non trivial element of N . The set of points x satisfying (exp h)F ss (x) = F ss (x) and h h x = id is denoted by H(h). (2) For every x in M , the elements h of N satisfying (exp h)F ss (x) = F ss (x) and h h x = id is denoted by ∆ 0

x . (3) If ∆ 0

x is non trivial, x is said to admit trivial holonomy. (4) the set of points of M admitting trivial holonomy is denoted by P.

Lemma 8. The set H(h) is F ss -invariant.
Proof. Let x be an element of H(h) and y an element in F ss (x). If y is sufficiently close to x, the loop ω h y considered in Remark 4 in order to define h h y can be chosen close to the loop ω h

x . These loops are then freely homotopic one to the other in F s (x). Hence we have h h y = h h x =id and y ∈ H(h). The case for general y in F ss (x) is obtained by iterating under exp(ta 0 ). Lemma 9. The subset P is either empty, or the entire M.

Proof. Let x be an element of P, i.e. an element of H(h) for some non-trivial element h. For every g in N, we have gx ∈ H(ad(g)h) hence P is N -invariant. It follows, together with Lemma 8, that P is F s -invariant.

Let now y be an element of F uu (x). Since the holonomy h h

x is trivial, if y is sufficiently close to x in F uu (x), the image (exp h)y lies in the same stable leaf F s (y) than y. Moreover φ h maps F ss (y) close to itself (since it preserves F ss (x)) and therefore we have F ss (y) = (exp h )F ss (y) for some element h close to h. The loop ω h y can be chosen so that it remains close to ω h

x ; hence to be the lifting in F s (y) of ω h x . Since h h x is trivial, it follows that h h y is also trivial, hence y ∈ H(h) ⊂ P.

Now for a general y in F uu (x), not necessarily close to x, some iterate exp(ta 0 )y will be close to exp(ta 0 )x in F uu (exp(ta 0 )x), hence in P anyway.

The set P is therefore invariant by the foliations F s and F u . The lemma follows.

From now on, we assume P = M.

Lemma 10. Let x 0 be an element of a compact φ-orbit O 0 . Then exp(∆ 0 x ) is a normal subgroup of the N -stabilizer ∆ x .

Proof. The key point is that the intersection between the strong stable leaf F ss (x 0 ) and O 0 is reduced to x 0 . Indeed, by the local product structure the intersection between O 0 and F ss δ (x) is reduced to x 0 , and the general claim follows by iterating under (the inverse of) a ξ-Anosov element fixing x 0 .

Therefore we have exp(∆ 0 x ) ⊂ ∆ x . It is straightforward to check that exp(∆ 0 x ) is a subgroup.

The fact that exp(∆ 0 x ) is normal in the stabilizer follows easily from the fact that the holonomy h adg(h) x is the conjugate under φ g of h h

x , hence trivial if h h x is trivial.

Let 0 = N r+1 N r ... N 1 = [N, N ] N 0 = N the lower central serie for N. For every x in Comp(φ), let k(x) be the biggest integer k for which exp(∆ 0

x ) ∩ N k is not trivial. Observe that k is N -invariant. We denote by ∆ 1

x the intersection exp(∆ 0 x ) ∩ N k(x) .

Lemma 11. For every x in Comp(φ) the weak unstable leaf F u (x) is fixed pointwise by ∆ 1 x .

Proof. Then, for every g in ∆ x and every γ in ∆ 1

x , the commutator gγg -1 γ -1 lies in exp(∆ 0

x ) ∩ N k+1 , hence is trivial. It follows that ∆ 1 x lies in the centralizer of ∆ x . Since N is nilpotent and ∆ x a lattice in it, ∆ 1

x is contained in the center Z of N (indeed, ∆ 1 x lies in the center of the Zariski closure of ∆ x , which is the entire N , see Theorem 2.10 in [START_REF] Raghunathan | Discrete subgroups of Lie groups[END_REF]).

It follows that for every h = exp(h) in ∆ 1

x , the entire orbit O(x) is fixed pointwise by h. Moreover, for every y in O(x), the unstable leaf F uu (y) is preserved, and fixed pointwise since the holonomy h h y is trivial. The lemma follows.

Lemma 12. Let x 1 and x 2 be two points in Comp(φ). If x 1 and x 2 belongs to the same basic set Λ i , then we have k(x 1 ) = k(x 2 ) and ∆ 1

x 1 = ∆ 1 x 2 .
Proof. Let y be a point in F u (x 1 ) ∩ F ss (x 2 ). Let h be an element of ∆ 0 x 1 whose projection by exp is in ∆ 1

x 1 . According to Lemma 11, y is a fixed point of exp(h), hence belongs to H(h). According to Lemma 8, x 2 lies in H(h). Hence, h lies in ∆ 0

x 2 . Moreover, since exp(h) ∈ N k(x 1 ) , we have:

k(x 2 ) ≥ k(x 1 ).
In a symetric way, one shows that every element in ∆ 0

x 2 with exponential in ∆ 1 x 2 is contained in ∆ 0

x 1 and that k(x 1 ) ≥ k(x 2 ). The lemma follows.

It follows that for every basic set Λ i , there is a non trivial discrete subgroup ∆ i of the center Z of N fixing Λ i pointwise. Actually, by Lemma 11, the entire

F u (Λ i ) is contained in Fix(∆ i ).
More precisely, there is an integer k i (the common value of k on Comp(φ) ∩ Λ i ) such that:

-no non trivial element of N k i +1 fixes pointwise F u (Λ i ) (indeed, such an element would be an element of ∆ 0

x ∩ N k(x)+1 for any x in Comp(φ) ∩ Λ i ), -∆ i is exactly the subgroup of N k i comprising the elements fixing pointwise F u (Λ i ), Moreover, ∆ i is contained in the center Z of N. It follows from these properties that if Λ j is any basic set contained in the closure of F u (Λ i ) then we have k j = k i and N k i = N k j . Since the graph G is connected (Definition 8, Lemma 7), all the basic sets have the same integer k i and the same N k i . It follows that elements of N k i acts trivially on the entire M. It is impossible if φ is faithful.

The proof of Proposition 6 is complete.

Actions of codimension one

In this section, we keep considering an Anosov action φ : N × M → M of a simply connected nilpotent Lie group N , but we know add the additional hypothesis:

E uu has dimension one.

4.1.

Affine structures along the strong unstable leaves. Most of the content of this section is classical. We refer to [10, section 4.1] for more details. An affine structure of class C 2 on R is equivalent to the data of differential 1-form on R, hence once fixed a differential 1-form ω 0 on M which does not vanish on E uu , any continuous function f : M → R induces an affine structure along leaves of F uu : the one induced by the restriction of f ω 0 on the leaf. We say that such an affine structure is φ-invariant if for every a ∈ N and every x ∈ M the restriction of φ a induces an affine diffeomorphism bewtween F uu (x) and F uu (φ a (x)). Proposition 7. There exists an unique affine structure along F uu depending continuously on the points and invariant by the action φ.

Proof. Let a 0 be an Anosov element of N expanding the leaves of F uu . Then, a contracting fixed-point Theorem on a suitable Banach space ensures the existence of a unique affine structure along F uu invariant by the one-parameter subgroup H 0 generated by a 0 (see [START_REF] Barbot | Géométrie transverse des flots d'Anosov[END_REF] or [24, Proposition 3.4.1, Remarque 3.4.2]). Since it is unique, this affine structure is also N (H 0 )-invariant. By an induction involving Lemma 1 we finally get that this affine structure is N -invariant.

For a proof of the completeness of the leaves, see [24, §1.3].

Corollary 2. Let a be an element of N , and x an element of M such that φ a (x) = x. Then the restriction of φ a to F uu (x) is differentially linearizable. In particular, either F uu (x) pointwise fixed by a, or x is an isolated fixed point of φ a in F uu (x). Proof. According to Lemma 3, such an Anosov action is a Nil-extension over a faithful Anosov action. Consider a compact orbit Λ\N : the holonomy of the stable leaf containing this orbit is linearizable (Corollary 2): it provides a morphism from Λ into R * . According to proposition 6, this morphism is injective. The theorem follows.

It follows that many results in [START_REF] Barbot | Transitivity of codimension one Anosov actions of R k on closed manifolds[END_REF], insensitive to Nil-extensions, still hold for Anosov actions of nilpotent Lie groups of codimension one, not necessarily abelian. More precisely:

Let π : M → M be the universal covering map of M and φ be the lift of φ on M . The foliations F ss , F uu , F s and F u lift to foliations F ss , F uu , F s and F u in M . We denote by Q φ be the orbit space of φ and π φ : M → Q φ be the canonical projection. We have the following properties:

• The orbits of φ are incompressible: every loop in a φ-orbit O which is homotopically non-trivial in O is homotopically non-trivial in M. • The foliations F uu , F ss , F u , F s and the foliation defined by φ are by closed planes.

The intersection between a leaf of F u and a leaf of F s is at most an orbit of φ. Every orbit of φ meets a leaf of F uu or F ss at most once.

• The universal covering of M is diffeomorphic to R n+k , and the orbit space Q φ is is homeomorphic to R n .

Last but not least: Theorem 6. Let φ be a codimension one Anosov action of a nilpotent Lie group N of dimension k on a closed manifold M of dimension n + k with n ≥ 3. Then, Ω(φ) = M .

Proof. This is a consequence of Theorem 1 in [START_REF] Barbot | Transitivity of codimension one Anosov actions of R k on closed manifolds[END_REF] and of the observation that a Nilextension of a topologically transitive action is still topologically transitive. However, we propose here another proof, assuming the theorem to be known for flows (cf. [START_REF] Verjovsky | Codimension one Anosov flows[END_REF]) and proving it in the case n ≥ 2.

According to Theorem 5 one can assume that the group N is R k with n ≥ 2. Then, since any subgroup of R is either cyclic or dense, it follows from Corollary 2 that any stable leaf F intersecting an unstable leaf F uu (θ 0 ) of a periodic orbit θ 0 but with θ 0 / ∈ F is locally dense; more precisely, F is contained in the interior of the closure F of F. On the other hand, it follows from Corollary 1 that every leaf of F s satisfies this property.

Let F be any leaf of F s , and G be a leaf in the closure F : this leaf is contained in the interior of its closure G, which is contained in F . Hence G is contained in the interior of F . Therefore, F is closed and open, i.e. the entire M. The Theorem then follows from Proposition 5. 4.3. Anosov actions of codimension two. In the previous section we have seen that if n ≥ 3, Anosov actions of codimension one are topologically transitive. In this section, elaborated with R. Varão, we consider the remaining case n = 2. We show: Theorem 7. Let φ be a codimension one Anosov action of a nilpotent Lie group N of dimension k on a closed manifold M of dimension k + 2. Then φ is a Nil-extension over an Anosov flow on a 3-manifold. This Theorem is a direct consequence of Theorem 5 and the main result of V. Arakawa in his (unpublished) Ph. D Thesis ( [START_REF] Arakawa | Sobre classificação de ações Anosov de R k em (k + 2)-variedades fechadas[END_REF]). However, we propose here a proof much shorter than the one appearing in [START_REF] Arakawa | Sobre classificação de ações Anosov de R k em (k + 2)-variedades fechadas[END_REF].

Proof. By a way of contradiction, assume that we have a codimension one Anosov action of a nilpotent Lie group N of dimension k on a closed manifold M of dimension k+2 that is not a Nil-extension over an Anosov flow on a 3-manifold. By Theorem 5 one can assume that N is R k with k ≥ 2. Let θ 0 be an element of the orbit space Q φ which is a lift of a compact orbit: θ 0 is fixed by a subgroup Λ of π 1 (M ) isomorphic to a lattice in R k , i.e. isomorphic to Z k . According to Corollary 2 and the local product structure near θ 0 , the action of Λ in Q φ near θ 0 is linearizable. More precisely: there is a neighborhood U of θ 0 in Q φ , morphisms ρ 1,2 : Λ →]0, +∞[ and a homeomorphism ψ : U → V ⊂ G s (θ 0 ) × G u (θ 0 ) ≈ R 2 such that: ∀γ ∈ Λ, ∀θ ∈ U ψ(γ.θ) = (e ρ 1 (γ) ψ 1 (θ), e ρ 2 (γ) ψ 2 (θ))

where ψ(θ) = (ψ 1 (θ), ψ 2 (θ)).

We denote by ρ(Γ) the subgroup of R 2 comprising elements of the form (ρ 1 (γ), ρ 2 (γ)) with γ ∈ Λ. We have two cases to consider:

(1) Either ρ(Γ) is discrete in R 2 (hence a lattice since k ≥ 2), (2) Or ρ(Γ) is not closed in R 2 .

In the first case, θ 0 lies in the closure of the orbit under Λ of any point in U \ {θ 0 }. In the second case, the orbit under Λ of any point θ in U \ {θ 0 } accumulates non-trivially to θ .

In both situations, we conclude that no element of U \ {θ 0 } can be a lift of a compact orbit of φ. It follows that the basic set containing the projection of θ 0 is reduced to one compact orbit.

Since θ 0 was arbitrary, we conclude that the nonwandering set Ω(φ) is a finite union of compact orbits. According to Corollary 1 the foliation F s has only finitely many leaves: contradiction.

Definition 4 .

 4 Let φ : G × M → M be an action of a simply connected Lie group G. The action is Nil-faithful if the induced action of N is faithful. Lemma 3. Every Nil-Anosov action of a simply connected Lie group is a Nil-extension over a Nil-faithful Nil-Anosov action of a simply connected Lie group.

  Recall that we have fixed a metric | • | on N = G, and than an Anosov subcone C is strict if nonzero elements of C are Anosov (Definition 3). Definition 6 (Nonwandering set of a strict Anosov subcone). A point x ∈ M is nonwandering with respect to a strict Anosov subcone C if for any open set U containing x there is an element a of C such that |a| ≥ 1 and φ exp(a) (U ) ∩ U = ∅. The set of all nonwandering points, with respect to C, is denoted by Ω(C).

Figure 1 .

 1 Figure 1. Iterations of local sections.

Figure 2 .

 2 Figure 2. A schematic view of the argument.

4. 2 .Theorem 5 .

 25 Anosov actions of codimension one are abelian. An important corollary of the previous section is: Every codimension one Anosov action of a nilpotent Lie group is a Nilextension over a codimension one Anosov action of R k .

  Remark 1. For any splitting ξ, let A ξ denote the set of ξ-Anosov elements. By the Lie product formula we have, for any a, b in A ξ : Consequently, sa + tb is an Anosov element in A ξ which preserves ξ. It follows that A ξ is a convex cone in G.Definition 3 (Anosov subcone). A subcone C is an open convex cone in G. A subcone isAnosov if it is made of Anosov elements. It is strict if any nonzero element of the closure C is Anosov.

	exp(sa + tb) = lim n→∞	exp(	sa n	) exp(	tb n	)	n , for any positive real numbers s, t.

  is a subalgebra. It follows immediatly that, for any cocompact lattices Γ of G the right action on Γ\G of the Lie group H admitting H as Lie algebra is Anosov. Moreover, if L 0 (h) is minimal (i.e. does not contain a proper Engel subalgebra) then it is a Cartan subalgebra (abrev. CSA) i.e. is nilpotent and equal to its own normalizer (see [28, Theorem page 80]). It happens -for example if G is real semisimple and L 0 (h) a hyperbolic CSA, see [11, Appendix A.3] -that L i (h) is trivial, then we obtain in this case an Anosov action of a nilpotent Lie group. More generally, one can construct from CSA's many examples of algebraic Anosov actions, i.e. actions on quotients Γ\G/K of Lie subalgebras H of G where K is a compact subgroup commuting with exp(H). In

The first author thanks the FAPESP and the French consulate in the state of São Paulo (Brazil) for the financial support grant "Chaires franco-brésiliennes dans l'état de São Paulo". The second author would like to thank CNPq of Brazil for financial support Grant 200464/2007-8.