Stéphanie Fleck
email: stephanie.fleck@univ-lorraine.fr

Gilles Simon
email: gilles.simon@loria.fr

An Augmented Reality Environment for Astronomy Learning in Elementary Grades: An Exploratory Study

Keywords: Astronomy, Augmented Reality, Learning, Inquirybased Sciences Education. ACM Classification H.5.1 Multimedia Information Systems

This paper describes an ongoing research comparing two 3D astronomical tangible models: an Augmented Reality model versus a physical model. According to IBSE principles, learners should investigate and manipulate in order to become conscious of the origin of astronomical phenomena, construct scientific knowledge and change their misconceptions. In primary French schools, physical models are usually used. However, children do not take advantage of these models and form new synthetic models instead of scientific ones. We aim at providing an adapted pedagogical environment support. An Augmented Reality environment was designed for inquiry-based learning. This tangible AR model shows augmented views of the celestial bodies and supports the pupils' investigations using spatial visual guides and views from a terrestrial observer. The AR model not only exposes the phenomena as in several Virtual Environments, but also allows pupils to virtually move the celestial bodies and test "as for real" their hypotheses. Our results show that the AR environment is particularly suitable for astronomy learning compared to the physical one. Only AR users have developed scientific conceptions of the explored astronomical phenomena and learnings have been significantly improved. Furthermore, we present some arguments in order to support the assumption that the AR model assists the process of scaffolding and motivation dynamic by enhancing task controllability and by promoting collaborative learning.

INTRODUCTION

Today, astronomy has to be taught at primary, secondary and high school as part of the curriculum. But very few targeted teaching material is available for teachers and learners in France (resources are mainly textbooks and physical models). So, there is a real need to develop digital astronomical didactic materials.

The specific project of our study concerns the development of a tangible user interface for an Augmented Reality (AR) environment adapted to Inquiry-Based Sciences Education (IBSE) [START_REF] Grandy | Reconsidering the Character and Role of Inquiry in School Science: Analysis of a Conference[END_REF][START_REF] Minner | Inquirybased science instruction-what is it and does it matter? Results from a research synthesis years 1984 to 2002[END_REF]. The main goal of our project is to provide easy-to-use teaching/learning tools for learners and teachers in elementary grades, in accordance with the French curriculum in astronomy. We aim at providing an adapted pedagogical support to learners to help them acquire new concepts and overtake misconceptions about fundamental astronomical concepts described in the primary grades curriculum. The main contribution of this study is a comparison between two 3D tangible models, physical and virtual, proposing the same possibilities of manipulation. We also evaluate the impact of AR on the development of children's knowledge on a complex astronomical concept (moon phases).

CHILDREN'S KNOWLEDGE OF ASTRONOMY AND AUGMENTED REALITY ENVIRONMENT

Some concepts of astronomy have been taught at elementary school in France since 1985. More precisely, the program is focused on the solar system, earth motion, day/night cycle, seasonal changes, moon motion and moon phases. For fifteen years, a lot of studies have shown that children of various cultures have difficulties in understanding contemporary scientific explanations of elementary astronomic phenomena [START_REF] Bryce | The Novice-Expert Continuum in Astronomy Knowledge[END_REF][START_REF] Hannust | Children's knowledge of astronomy and its change in the course of learning[END_REF][START_REF] Hannust | Young children's acquisition of knowledge about the Earth: A longitudinal study[END_REF][START_REF] Nobes | The development of scientific knowledge of the Earth[END_REF][START_REF] Straatemeier | Children's knowledge of the earth: new methodoligical and statistical appraoch[END_REF][START_REF] Vosniadou | Mental models of the earth: A study of conceptual change in childhood[END_REF][START_REF] Vosniadou | Mental Models of the Day/Night Cycle[END_REF]. Astronomy is a field where the information collected by means of personal experience contradicts contemporary scientific theory [START_REF] Hannust | Children's knowledge of astronomy and its change in the course of learning[END_REF].

From a cognitive point of view, children synthesize (construct) their knowledge of the world and the universe on the basis of two information sources: observations of the world and explanations given by other people [START_REF] Vosniadou | Mental models of the earth: A study of conceptual change in childhood[END_REF]. In the continuity of Bachelard [START_REF] Bachelard | La formation de l'esprit scientifique. Contribution à une psychanalyse de la connaissance objective[END_REF], Vosniadou and her colleagues [START_REF] Vosniadou | Exploring the relationships between conceptual change and intentional learning[END_REF][START_REF] Vosniadou | Mental models of the earth: A study of conceptual change in childhood[END_REF][START_REF] Vosniadou | Mental Models of the Day/Night Cycle[END_REF][START_REF] Vosniadou | Designing learning environments to promote conceptual change in science[END_REF][START_REF] Vosniadou | Reconsidering the role of artifacts in reasoning: Children's understanding of the globe as a model of the Earth[END_REF] have determined three types of cognitive models, i.e., three steps in the construction of scientific mental models of the world and the universe:

1. Initial model: the initial model is based on children's everyday experiences (e.g., seeing the flat surface of the earth) and on the entrenched beliefs derived from those early experiences; 2. Synthetic model: children form synthetic models after they are exposed to the contemporary scientific information. These are formed as a result of attempts to reconcile their presuppositions with the information they receive from adults; 3. Scientific model: since adolescence, the majority of Western people have accepted as truth the scientific views. Children learn informally about the physical and astronomical phenomena in daily life long before official instruction are provided at schools. This can be a source of problems, especially when children's preliminary knowledge radically differs from what is taught in school.

If these knowledge are integrated and consistently used, children tend to reinterpret the new information in accordance with their preliminary models and to form new synthetic models instead of scientific ones [START_REF] Hannust | Children's knowledge of astronomy and its change in the course of learning[END_REF][START_REF] Hannust | Young children's acquisition of knowledge about the Earth: A longitudinal study[END_REF][START_REF] Vosniadou | Reconsidering the role of artifacts in reasoning: Children's understanding of the globe as a model of the Earth[END_REF]. As recommended by European and French institutions [START_REF] Forsthuber | Science Education in Europe: National Policies, Practices and Research[END_REF][START_REF]France -Ministère de l'éducation nationale et ministère de l'enseignement supérieur et de la recherche[END_REF] and to make science learning more efficient [START_REF] Duschl | Taking Science to School : Learning and Teaching Science in Grades K-8[END_REF][START_REF] Fleck | Tâtonnement expérimental : Processus naturel d'apprentissage[END_REF][START_REF] Grandy | Reconsidering the Character and Role of Inquiry in School Science: Analysis of a Conference[END_REF][START_REF] Minner | Inquirybased science instruction-what is it and does it matter? Results from a research synthesis years 1984 to 2002[END_REF], especially in astronomy [START_REF] Taber | International Handbook of Research on Conceptual Change[END_REF][START_REF] Vosniadou | Reconsidering the role of artifacts in reasoning: Children's understanding of the globe as a model of the Earth[END_REF] the following inquiry-based pedagogical principles [START_REF] Saltiel | POLLEN -L'enseignement des sciences fondé sur l'investigation. Conseils pour les enseignants. La main à la pâte[END_REF] must be proposed: (1) explicitly talking about children's preliminary knowledge and exploring different ideas to make them become conscious of their understanding/expertise; [START_REF] Auzias | 50 activités avec l'astronomie à l'école cycle 3 et en 6e[END_REF] showing the inconsistencies between everyday and scientific explanations and their reasons; (3) making children become conscious of their entrenched beliefs in order to change them; (4) giving the new explanations verbally and allowing pupils time to consolidate their thinking; (5) facilitating debates and leaving enough time for discussions. We argue that AR environments are particularly suitable for astronomy learning according to IBSE precepts. AR is one of the most promising technologies in education for about ten years [START_REF] Martin | New technology trends in education: seven years of forecasts and convergence[END_REF]. However, implementations in primary school are uncommon (e.g. [START_REF] Campos | Fostering collaboration in kindergarten through an augmented reality game[END_REF]8,[START_REF] Kerawalla | Making it real": Exploring the potential of Augmented Reality for teaching primary school science[END_REF][START_REF] Luckin | Limitless or pointless?: An Evaluation of Augmented Reality in the School and Home[END_REF]). Existing works mainly focus on applications in university education (architecture, biochemistry, mathematics, anatomy or physiology…) (e.g. [START_REF] Chen | A study of comparing the use of augmented reality and physical models in the chemistry education[END_REF][START_REF] Di Serio | Impact of an Augmented Reality System on Students' Motivation for a Visual Art Course[END_REF][START_REF] Kaufmann | Mathematics and geometry education with collaborative augmented reality[END_REF][START_REF] Wojciechowski | Evaluation of learners' attitude toward learning in ARIES augmented reality environments[END_REF]). To our knowledge, only one study concerns astronomy [START_REF] Shelton | Using augmented reality for teaching Earth-Sun relationships to undergraduate geography students[END_REF]. However, one of the most promising development axes probably emanate from the scholar public. Moreover, the educational subjects that mobilize the appropriation of abstract concepts and spatial representations are very frequent in primary school programs, and all those subjects could be supported by AR. Some prototypes of Virtual Reality (VR) environments have been proposed to teach/learn astronomy (e.g., [START_REF] Bakas | Design of virtual environments for the comprehension of planetary phenomena based on students' ideas[END_REF][START_REF] Gazit | Emerging Conceptual Understanding of Complex Astronomical Phenomena by Using a Virtual Solar System[END_REF]) but these are too complex for elementary learners, or too expensive for French teachers, or not adapted to the French curriculum in astronomy for elementary levels. Moreover, they do not open the possibility for real manipulations. The AR tools seem to be the best environments as feedback provider and simulator for numerous teachers [START_REF] Salmi | Towards an Open Learning Environment via Augmented Reality (AR): Visualising the Invisible in Science Centres and Schools for Teacher Education[END_REF]. In comparison to VR, which aims at immersing the user in a synthetic environment, AR is a technology that allows computer-generated virtual imagery information to be overlaid onto a live direct or indirect real-world environment in real time [START_REF] Azuma | A survey of augmented reality[END_REF][START_REF] Lee | Augmented Reality in Education and Training[END_REF][START_REF] Simon | Intégrer images réelles et images 3D -Post-production et réalité augmentée[END_REF]. In other words, in AR environments both virtual and real objects can co-exist and interact in real time. The user experiences are more realistic and complex spatial relationship can be easily visualized [START_REF] Arvanitis | Human factors and qualitative pedagogical evaluation of a mobile augmented reality system for science education used by learners with physical disabilitie[END_REF].

In order to get a correct alignment between the virtual scene and the real scene, the pose of the observer (or camera) with respect to the scene has to be known at all times. A widely-used method to get the pose in real-time consists in using artificial markers that can be rapidly and efficiently detected and identified in a video flow. Several implementations are available to perform detection, we use Artoolkit [START_REF] Kato | Marker tracking and HMD calibration for a video-based augmented reality conferencing system[END_REF] in our experiments. The term "tangible interface" is used to designate interfaces that allow manual manipulations of the markers, where each marker is associated with a 3D object or a particular action. This kind of interface provides "sensorimotor feedback" [START_REF] Shelton | Exploring a cognitive basis for learning spatial relationships with augmented reality[END_REF]. Direct manipulations can supplement the deficiency of mouse-based computer-generated visualisation, since mouse manipulation is an indirect physical manipulation [START_REF] Chen | A study of comparing the use of augmented reality and physical models in the chemistry education[END_REF][START_REF] Shelton | Exploring a cognitive basis for learning spatial relationships with augmented reality[END_REF]. According to Shelton and Hedley [START_REF] Shelton | Using augmented reality for teaching Earth-Sun relationships to undergraduate geography students[END_REF], AR interfaces do not change the delivery of instruction content but change the way that content is understood, through a unique combination of visual and sensory information.

PRESENTATION OF THE AR LEARNING ENVIRONMENT

A virtual learning environment has been designed to enhance the conceptualization of earth and moon shadows' origin, the length of the day and the night on the earth and on the moon, and finally the evolution of moon phases. The AR model does not just expose the phenomena as in several virtual environments (e.g. [START_REF] Bakas | Design of virtual environments for the comprehension of planetary phenomena based on students' ideas[END_REF][START_REF] Gazit | Emerging Conceptual Understanding of Complex Astronomical Phenomena by Using a Virtual Solar System[END_REF]). Here, the virtual celestial objects can be moved to make "as for real". In our AR environments, virtual sun, earth and moon are associated to specific patterns. Therefore each marker is easily identifiable. Visual guides are provided to help the user understand the three celestial bodies' relationship. Sun, earth and moon appear realistic as they are represented using textured 3D spheres (textures were obtained from space images).

The light properties were taken into account and self shadows of earth and moon were directly produced by an omnidirectional light source associated to the sun. Different visual guides are proposed to support learners (see Figure 1). In particular, an optional vignette can be displayed to see the subjective view of a virtual terrestrial observer in real-time.

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1-The moon marker is put in front of the camera ; as the sun marker has not been introduced yet, the virtual moon is completely black.

-The sun marker is introduced

and the virtual moon is illuminated by it. The sun marker is moved, changing the appearance of the moon as observed "from space" (camera view).

3-The sun marker is removed but the virtual sun is still displayed; as long as the sun marker is hidden, the last measured position of the sun relative to the moon is used to display the sun.

4-The moon marker is moved in order to observe the moon from different viewpoints. As the position of the sun relative to the moon is fixed, the moon illumination is not changed during this operation.

5-All markers (moon, sun and earth) are put on the table. A virtual observer is drawn on the surface of the earth (in France by default); a vignette is displayed bottom right of the screen, showing the moon's appearance for the terrestrial observer.

6-The terrestrial observer has been moved toward the east along a geodesic path, using a keyboard key;; the moon's appearance has changed in the vignette; moreover, as the terrestrial observer is now on the dark side of the earth, a night sky is displayed in the vignette.

7 -The sun marker is moved, changing the appearances of the celestial bodies, both for the spatial and the terrestrial views Therefore, this AR environment provides users with experiences they would otherwise not be able to experience in the physical world.

Moreover, it is possible to adjust the difficulty level according to education objectives, to adapt the level of freedom to move markers. Then, this AR model can be used in inquiry-based education. It is certainly the first Augmented Reality astronomical model adapted to primary students, IBSE and French institutional curriculum. Traditionally, the teachers of French primary classes use classical physical models which possess a large part of these parameters (i.e. manipulation possibilities, physical representation of celestial bodies, investigation possibilities...) crucial for astronomy learning [START_REF] Minner | Inquirybased science instruction-what is it and does it matter? Results from a research synthesis years 1984 to 2002[END_REF][START_REF] Padalkar | Designed and Spontaneous Gestures in Elementary Astronomy Education[END_REF][START_REF] Rolando | Astronomie à l'école élémentaire: quelques réfléxions sur la construction des compétences[END_REF][START_REF] Saltiel | POLLEN -L'enseignement des sciences fondé sur l'investigation. Conseils pour les enseignants. La main à la pâte[END_REF].

The aim of this study is to assess whether this AR environment improves inquiry-based learning compared to a physical model.

METHOD

With an exploratory experiment conducted with 39 young French learners from primary classes, we investigated the potential of this AR environment to improve learning in astronomy by comparison with a "traditional" physical model.

Participants

All the learners were recruited from Grades 4 and 5 in one French school. Students are from 8 to 11 years old and 59% of them are male. The pre-assessment on moon phases' origins pre-conceptions has allowed subdividing the participants in two similar panels (p-value= 0.8269 using Fisher's Exact Test) also taking into account the age and gender of pupils (see Table 1). The pupils' experiences on both models have been included in an inquiry-based pedagogical sequence, consisting of 6 steps synthesized in Table 2.

Pupils were separately assigned to panels at the fourth and fifth steps of the pedagogical sequence. The first panel -panel A-(see Table 1) had access to the AR model to resolve astronomical problems (see Figure 1) whereas the second panel -panel B-had access to a classical physical model to resolve exactly the same problems (see Figure 3). This physical model was constructed and implemented as described in some manuals commonly used by French teachers (e.g. [START_REF] Auzias | 50 activités avec l'astronomie à l'école cycle 3 et en 6e[END_REF][START_REF]France -Ministère de l'Éducation nationale de la Jeunesse et de la Vie associative[END_REF]). Both panels used the same inquiry-oriented pedagogical sequence during the other steps.

Setting and procedure

As exposed by Campos, Pessanha and Jorge [START_REF] Campos | Fostering collaboration in kindergarten through an augmented reality game[END_REF], the impact of technology cannot be fully understood without considering the global educational context. The study took place during 6 non-consecutive days, in respect with parental and institutional agreements. The research team conducted all steps of the pedagogical sequence. So, a combination of pre and post-tests, analyses of pupils' workbooks, digital logs of the markers for the AR model, observations and videotaping, could develop a better understanding of the influence of these environments on learnings.

Steps 2 and 3 aimed at assessing and talking about the children's preliminary knowledge about origin of moon phases observed during the previous month (step 1). The discovery time, due to the AR "magical effect", may limit the interest in learning as in the Kerawalla, Luckin, Seljeflot and Woolard [START_REF] Kerawalla | Making it real": Exploring the potential of Augmented Reality for teaching primary school science[END_REF] study. Other AR environments were presented during step 3 to the children of panel A. They could play, test marker mobility, understand the feedback nature and then adapt to this type of augmented environment.

During steps 4 and 5, the AR system was set up in computer rooms to perform the exercises and students were videotaped in action. A Logitech QuickCam pro 9000 video camera was connected to a laptop computer Centrino2 running Windows Vista. Simultaneously, the classical physical model was set up in another small room of the school, where the overhead lights could be switched off to darken the room during the whole experimentation (see Figure 2). In both conditions ("AR environment" vs. "Physical model") pupils were asked to cast doubt on their conceptions. Students of panel A filtered through the AR model over a day at step 4 and with the same methods one week later for the step 5.

Simultaneously, panel B filtered through the classical physic model. Both had to investigate on the origin of moon phases based on similar problems.

Table 2: Pedagogical sequence, similar for both panels (except for parts in italics), including time for investigations (steps 4 and 5) using either models types (AR or physical models).

First, they had to find the principal moon phases viewed from earth (new and full moons, crescents, gibbous and quarters) and testing their conceptions on the moon shape evolution observed during the previous month (e.g. moon has white and black hemispheres, the dark zone on moon is due to earth's shadow or clouds, the evolution of the way we see the own shadow of moon during the lunar revolution). These experiments lasted 45' on average for each, working by pairs for both panels. Students had their experiment workbooks to note their hypotheses and results.

Secondary, they had to be able to replace the moon in the sun/earth/moon system viewed from space during 4 moon phases (new and full moons, waxing crescent, last quarter) by testing their conceptions.

Step

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1-A small ball (moon) is placed against the light (sun). It is moved around the lamp in order to observe the moon from different viewpoints. This is then a modelling of the shadow production on moon and of the apparent moon phases observed from earth (as in picture 2, 3 and 4-Figure 1)

2-The moon-ball and another big one (earth) are placed against the light. The moon ball is moved around the earth ball in order to observe the moon shapes evolutions. It is a modelling of the lunar month view from space (as in pictures 5, 6 and 7 -Figure 1). Both experiments for the two panels were videotaped. The marker motions were stored during each working session to provide statistical data. During step 6, the knowledge of pupils on astronomical concepts were once again assessed and each of them was asked to put on paper the astronomical concepts constructed during this sequence.

RESULTS AND DISCUSSION

In this paper, data analysis is mainly focused on pre-and post-assessments. These assessments are associated to verbal exchanges noted during steps 2&6 and to analyses on pupil workbooks. Both correspond to question series about moon's phases (names, shape, origins, and physical explanation) constructed after Hannust and Kikas [START_REF] Hannust | Children's knowledge of astronomy and its change in the course of learning[END_REF], Straatemeier, van der Maas and Jansen [START_REF] Straatemeier | Children's knowledge of the earth: new methodoligical and statistical appraoch[END_REF], Thouin [START_REF] Thouin | Enseigner les sciences et la technologie au préscolaire et au primaire[END_REF], and an exercise from a classical school manual [START_REF] Auzias | 50 activités avec l'astronomie à l'école cycle 3 et en 6e[END_REF]. From 4 different earth/moon/sun locations, seen from above, looking down on the Earth's North Pole, students had to draw the 4 different moon phases as viewed by a terrestrial observer. Both assessments verified the skills "Explain the moon shape origin and the evolution of moon phases" and "Recognise and represent exactly the apparent moon phases for a terrestrial observer from spatial Earth/Moon/Sun representations".

As reported by Hannust and Kikas [START_REF] Hannust | Young children's acquisition of knowledge about the Earth: A longitudinal study[END_REF], for any learning, children need new terms, facts and explanations. Then, further knowledge had to be introduced in order to explain the phenomena.

The pre-assessment and post-assessment were analysed predominantly using 3 criteria, indicators of student learning: (1) type of vocabulary (scientific or not) used both verbally and in writing form; (2) drawing task quality (e.g. construct using observation, using didactical routine...); (3) explanations and arguments provided to open-ended questions.

The pre-assessment worksheets were collected during step 2. Results (see Figure 3 and Table 3) indicate that the majority (56%) of students explain lunar month using naive mental models (e.g. "moon has white and black hemispheres", "dark sector on moon is due to projected earth shadow") or entrenched beliefs (e.g "the moon is not entirely visible because of clouds between earth and moon", "it is due to the shadow of another planet"). Only 7 of the 39 students had a scientific model to explain and no difficulties to draw moon phases during the exercises. These results then confirmed the numerous recent studies on this subject [START_REF] Hannust | Children's knowledge of astronomy and its change in the course of learning[END_REF][START_REF] Hannust | Young children's acquisition of knowledge about the Earth: A longitudinal study[END_REF][START_REF] Nobes | The development of scientific knowledge of the Earth[END_REF][START_REF] Panagiotaki | Children's representations of the earth: A methodological comparison[END_REF][START_REF] Straatemeier | Children's knowledge of the earth: new methodoligical and statistical appraoch[END_REF]. The post-assessment were collected during step 6, one week after the last investigations on models, and after the 5 steps of learning.

For the two different panels, learning significantly increased (p-value = 0.008 for panel A and p-value = 0.020 for panel B, using Fisher's exact test). Most of the pupils make sufficient progress in learning to succeed, in whole or in part, the post-test (see Figure 3 and Table 3: "Acquired" and "Underway").

For both panels, we did not identify remaining entrenched beliefs and only 5 students continued to explain lunar month using naive mental models, and had many difficulties to draw moon phases ("Not acquired"). Synthetic models were mostly expressed (e.g. "it depended on how we saw the dark part of the moon") However, and even if students were in the same pedagogical conditions (see Table 2), the number of pupils who present the astronomical skills required ("Acquired") significantly improved when students investigated the AR model (p-value = 0.019 using Fisher's exact test) compared to the physical one (p-value = 0.935 using Fisher's exact test). Moreover, their scientific expression is more pronounced (45% of the panel A compared with 26% of the panel B) and even for the younger pupils (CE2 -8/9 years old).

These results confirm that when using physical models, users do not much benefit from manipulations to change their conceptions [START_REF] Martinand | Enseignement et apprentissage de la modélisation en sciences[END_REF][START_REF] Rolando | Astronomie à l'école élémentaire: quelques réfléxions sur la construction des compétences[END_REF].

Secondly, we used pedagogical observations and videotaping to identify the factors that might influence improvements and if it has an impact on specific learning outcomes.

During step 4, students of panel A were more motivated and eager to go on. This was due to the attractiveness of the AR model. These behaviours were limited in panel B. Nevertheless, this eagerness rapidly decreased. Thus, free manipulations observed during the first part of step 4-investigation were not observed during the second part and during step 5. For these young pupils, currently exposed to new technologies, and after this appropriation time, it is not the attractiveness of new technology that motivates users but really the scientific question and its challenge goal. Therefore, this study based on a learning sequence (and not only on one working session) indicates the importance of taking into account this appropriation time before inferring on influences due to a new virtual environment on user interactions, learning or motivation.

In both modelling contexts (steps 4 and 5) the duration of student activities were mainly the same (35'), all started quickly their investigations and did not hesitate to request teacher (about 6 to 7 times per inquiry-based sessions). All pairs of each panel, in exception of GR14-Panel A and GR7-Panel B, succeeded in resolving the astronomical problems after testing their various hypotheses and discussing together their protocols and results. Both models improved student investigation and were adapted to the age of the learners (8 to 11 years old).

However, videotaping qualitative analyses indicate two major differences between the two panels which could 6 6

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influence learning improvements associated to the AR model:

1-Tool for scaffolding [START_REF] Vygotski | Pensée et langage[END_REF]. Every physical model users has shown up many handling difficulties, particularly to move balls, to decide to turn around the table, to orient the devices etc. But all these movements were yet necessary to construct a correct spatial understanding with this type of models. Combined abstract thinking, cognitive and motor efforts for spatial situation involve here too much solicitations. For the AR model, users encountered problems with intermittent images when their hand or fingers occluded parts of the marker, and had some difficulties in understanding the marker positions and orientations in the webcam view.

Most of the students of panel B (except the GR5-Panel B pair) indicated recurrent visual difficulties during their manipulations. For example, to recognize the moon position between earth and sun and be able to explain a waxing crescent, viewed by a terrestrial observer, students had to orient their eyes as if they were the terrestrial observer and had to decentre themselves. Inversely, the users of the AR model systematically used the vignette to verify their hypotheses. As expected, realtime view of the terrestrial observer facilitated and supported users' investigations. The realism and the simple mobility of the virtual celestial objects were strongly promoted. Moreover, by simply manipulating markers, students were able to easily view and interact with complex phenomena. Also, the virtual model provides the necessary visual and tactile information needed to understand the optical phenomena at the origin of moon phases and then develop their conceptions. In other words, the AR model is an important tool for scaffolding pupils on this particular point. 2-Dynamic of motivation support [START_REF] Viau | La motivation en contexte scolaire[END_REF]. The step 5 of the panel B, students frequently dropped out their investigations (more than twice more than in panel A groups). As we have seen above, the AR environment enhances usability and plays an important role in scaffolding learning. Then, the AR model allowed users to concentrate on their tasks and then enhanced task controllability. Moreover, in both models, students worked together but not with the same behaviour. Dillenbourg, Baker, Blaye and O'Malley [START_REF] Dillenbourg | The evolution of research on collaborative learning. Learning in Humans and Machine: Towards an interdisciplinary learning science[END_REF], or Panitz [START_REF] Panitz | Collaborative versus cooperative learning: A comparison of the two concepts which will help us understand the underlying nature of interactive learning[END_REF], indicate that collaboration involves joint intention, mutual and coordinate effort, whereas cooperation involves greater labour division.

In panel B, using the physical model, students systematically divided up tasks and proceeded in interaction. For example, a student gave movement indications to a second one, his assistante.g."Put the moon here... Wait! I look", "turn around"-. They were complementary but organized in separate tasks. In other words they have cooperated. This is certainly due to the number of tasks, movements and cognitive efforts exposed previously. Dissensions were frequent in these groups and in most cases, one directed the investigations and moreover, only one is in good position for observations. In panel A, they frequently exchanged their points of view and ideas, worked together towards the same purpose (e.g. one moved moon markers or terrestrial observer and the second provided screen control) and manipulated the system alternatively. Twothirds of panel A-groups assisted each-other. These collaborative work, based on mutual aid and associated work, was preferred by students.

The AR model seems to facilitate collaborative learning interactions between students. The AR model enhances social context and environment, which are important parts of student motivation and self-regulation.

CONCLUSIONS AND FUTURE WORK

This study, based on an inquiry-based learning sequence, and not only on one working session, indicates the importance of taking into account the need for appropriation time before inferring the influence of a new virtual support. It shows the relevance of an AR model compared to a physical one to provide a learning environment easy to use for students in grades 4 and 5 and adapted to IBSE. This augmented learning environment enhances significantly astronomical learning. Further assessment analyses, qualitative analyses of combined data of videotaping, pupils' workbooks and observations improve our comprehension of the AR environment rule on learning. Because the AR model enhances sensory motor interactions, visual guides and realistic representations, pupils benefit from a reduced complexity to construct a scientific comprehension of phenomena. Therefore, AR model assists the process of scaffolding. Then, this AR learning environment contributes to make the misconceptions became conscious in order to construct scientific mental models. Moreover, cognitive support and handling ability, improve their capacity in resolving astronomical problems. The AR model then allowed users to concentrate on their tasks and then enhanced task controllability. By promoting collaborative learning interactions rather than cooperative ones, the AR model takes part of the dynamic motivational process.

The next phase of this research will take spatial procedures into specific account, through statistical data of marker motions, used during investigations with the AR model. Related to the assessments and combined data, we aim at providing arguments to characterize the procedures used by children to overtake their conceptions. Increasing the participant number would improve our inferences and would enhance comprehension on inquiry-based learning strategies.

Additionally, this experiment will be duplicated on preservice teacher students. As children, currently teachers (i.e., adults) have principally non-scientific conceptions about the astronomical notions they will have to teach. We aim at characterizing if adults overtake misconceptions as do children.

7 7

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Figure 1 :

 1 Figure 1 : An example use of the AR learning environment. First person perspective. Visual guides proposed to support learners :(a) dashed-lines between bodies' centres (b) a terrestrial observer (c) vignette showing the terrestrial observer's view in real-time (optionally proposed).

Figure 2 :

 2 Figure 2 : An example use of the physical model traditionally used in primary school. (a) Symbolic representation of a terrestrial observer.

Figure 3 :Table 3 :

 33 Figure 3: Assessment results summarized in acquisition levels.

Table 1 : Subdivision of the 39 pupils in the two panels. CM2 corresponding to 10/11 years old students, CM1 to 9/10 years old students and CE2 to 8/9 years old students.

 1

	PANEL		A	B
	n		20	19
	Male nbr (%)		12 (60%)	11 (58%)
	Age Mean (SD)		9,65 (1,03) 9 ,31 (1,15)
		CE2	7	7
	French Grade	CM1	4	3
		CM2	9	9
		Acquired	3	4
	Pre-assessment results	Underway	6	4
		Not Acquired	11	11

ACKNOWLEDGEMENTS

The authors thank the school, their teachers and pupils who took part in this study. Thanks to Yves Medri, Beatrice Leonori, Jerome Dinet and Christian Bastien for their technical and personal supports.


Abed Bounemoura

Vadim Kaloshin
email: vadim.kaloshin@gmail.com

Generic fast diffusion for a class of non-convex Hamiltonians with two degrees of freedom

In this paper, we study small perturbations of a class of non-convex integrable Hamiltonians with two degrees of freedom, and we prove a result of diffusion for an open and dense set of perturbations, with an optimal time of diffusion which grows linearly with respect to the inverse of the size of the perturbation.

Introduction and statement of the result 1.Introduction

In this paper, we consider small perturbations of integrable Hamiltonian systems which are defined by a Hamiltonian function of the form H(θ, I) = h(I) + εf (θ, I), (θ, I) ∈ T n × R n , 0 ≤ ε < 1, where n ≥ 2 is an integer and T n = R n /Z n . When ε = 0, H = h is integrable in the sense that the action variables I(t) of all solutions (θ(t), I(t)) of the system associated to h are first integrals, I(t) = I(0) for all times t ∈ R. The sets I = I 0 , for I 0 ∈ R n , are thus invariant tori of dimension n in the phase space T n × R n , which moreover carry quasi-periodic motions with frequency ω(I 0 ) = ∇h(I 0 ), that is θ(t) = θ(0) + tω(I 0 ) modulo Z n . From now on we will assume that the small parameter ε is non-zero, in which case the system defined by H can be considered as an ε-perturbation of the integrable system defined by h.

In the sixties, Arnold conjectured that for a generic h, the following phenomenon should occur: "for any points I ′ and I ′′ on the connected level hypersurface of h in the action space there exist orbits connecting an arbitrary small neighbourhood of the torus I = I ′ with an arbitrary small neighbourhood of the torus I = I ′′ , provided that ε is sufficiently small and that f is generic" (see [START_REF]Mathematical problems in classical physics[END_REF]). This is a strong form of instability. A weaker form of this conjecture would be to ask for the existence of orbits for which the variation of the actions is of order one, that is bounded from below independently of ε for all ε sufficiently small. To support his conjecture, Arnold gave an example in [START_REF]Instability of dynamical systems with several degrees of freedom[END_REF] where this weaker form of instability is satisfied, with n = 2, h convex and f a specific time-periodic perturbation (so this is equivalent to n = 3, h quasi-convex and f a specific time-independent perturbation). The phenomenon highlighted in [START_REF]Instability of dynamical systems with several degrees of freedom[END_REF] is now known as Arnold diffusion.

KAM stability

Obstructions to Arnold diffusion, and to any form of instability in general, are widely known following the works of Kolmogorov and Arnold on the one hand, and the work of Nekhoroshev on the other hand. In [START_REF] Kolmogorov | On the preservation of conditionally periodic motions for a small change in Hamilton's function[END_REF], Kolmogorov proved that for a non-degenerate h and for all f , the system defined by H still has many invariant tori, provided it is analytic and ε is small enough. What he showed is that among the set of unperturbed invariant tori, there is a subset of positive measure (the complement of which has a measure going to zero when ε goes to zero) who survives any sufficiently small perturbation, the tori being only slightly deformed. The non-degeneracy assumption on h is that at all points, the determinant of its Hessian matrix ∇ 2 h(I) is non-zero. Then, under a different non-degeneracy assumption on h, namely that the determinant of the square matrix ∇ 2 h(I) t ∇h(I) ∇h(I) 0 is non-zero at all points, Arnold proved in [START_REF] Arnol'd | Proof of a theorem of A.N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations[END_REF], [START_REF]Small denominators and problems of stability of motion in classical and celestial mechanics[END_REF] a similar statement but with a set of tori inside a fixed level hypersurface. In particular, for n = 2, a level hypersurface is 3-dimensional and the complement of the set of invariant 2-dimensional tori is disconnected, and each connected component is bounded with a diameter going to zero as ε goes to zero.

As a consequence, it can be proved more precisely that for n = 2 and if h is non-degenerate in the sense of Arnold, along all solutions we have

|I(t) -I(0)| ≤ c √ ε, t ∈ R,
for some positive constant c. Therefore we have stability for all solutions and for all time. Now for any n ≥ 2, and if h is either Kolmogorov or Arnold non-degenerate, we have perpetual stability only for most solutions, those lying on invariant tori, and Arnold's example shows that this cannot be true for all solutions. The consequence of these results is that Arnold diffusion cannot exist for n = 2 if h is Arnold non-degenerate, and for n ≥ 2 and h Kolmogorov or Arnold non-degenerate, the unstable solution, if it exists, must live in a set of relatively small measure.

Nekhoroshev stability

In an other direction, in the seventies Nekhoroshev proved ([Nek77], [START_REF]An exponential estimate of the time of stability of nearly integrable Hamiltonian systems II[END_REF]) that for any n ≥ 2, for a non-degenerate h and for all f , along all solutions we have

|I(t) -I(0)| ≤ c 1 ε b , |t| ≤ exp c 2 ε -a ,
for some positive constant c 1 , c 2 , a and b, provided ε is small enough and the system is analytic. So solutions which do not lie on invariant tori are stable not for all time, but during an interval of time which is exponentially long with respect to some power of the inverse of ε.

The consequence on Arnold diffusion is that the time of diffusion, that is the time it takes for the action variables to drift independently of ε, is exponentially large. The integrable systems non-degenerate in the sense of Nekhoroshev, which are called steep, were originally quite complicated to define, but an equivalent definition was found in [START_REF] Ilyashenko | A steepness test for analytic functions[END_REF] and [START_REF] Niederman | Hamiltonian stability and subanalytic geometry[END_REF]: h is steep if and only if its restriction to any affine subspace has only isolated critical points. Such functions can be proved to be generic in a rather strong sense ( [START_REF] Nekhoroshev | Stable lower estimates for smooth mappings and the gradients of smooth functions[END_REF]), and the simplest (and also steepest) functions are the convex or quasi-convex ones (convex or quasi-convex functions are those for which the stability exponent a in Nekhoroshev estimates is the best). Note that convex (respectively quasi-convex) functions are Kolmogorov (respectively Arnold) non-degenerate. So the results of Kolmorogov, Arnold and Nekhoroshev restrict the possibility of diffusion, both in space and in time, at least provided the corresponding non-degeneracy assumptions are met.

Arnold's example and "a priori" unstable systems

Following the original insight of Arnold in [START_REF]Instability of dynamical systems with several degrees of freedom[END_REF], much study have been devoted to perturbations of a special class of Hamiltonian systems, which are called "a priori" unstable, where these restrictions are much less stringent. We won't try to give a precise definition of "a priori" unstable systems, but these systems are integrable in the larger sense of symplectic geometry (they have n first integrals in involution and independent almost everywhere) but display hyperbolic features (typically they have a normally hyperbolic invariant manifold), and by opposition, the systems we are considering are called "a priori" stable. These simpler "a priori" unstable systems are now well-understood, and many results confirm that instability occurs for a generic perturbation, see for instance [START_REF] Treschev | Evolution of slow variables in a priori unstable Hamiltonian systems[END_REF], [START_REF] Cheng | Existence of diffusion orbits in a priori unstable Hamiltonian systems[END_REF], [START_REF] Delshams | A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model[END_REF], [START_REF] Gidea | Shadowing orbits for transition chains of invariant tori alternating with Birkhoff zones of instability[END_REF], [START_REF] Bernard | The dynamics of pseudographs in convex Hamiltonian systems[END_REF], [START_REF] Delshams | Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems[END_REF], [START_REF]Arnold diffusion in Hamiltonian systems: a priori unstable case[END_REF], and [START_REF]Obstruction argument for transition chains of tori interspersed with gaps[END_REF].

"A priori" stable systems

The situation for "a priori" stable systems is much more complicated. In [START_REF] Mather | Arnold diffusion I : Announcement of results[END_REF] (see also [START_REF]Arnold diffusion by variational methods, Essays in mathematics and its applications[END_REF] for a recent corrected version), Mather announced a proof of Arnold conjecture in a special case, that is a strong form of Arnold diffusion for a generic time-dependent perturbation of a convex integrable Hamiltonian with n = 2 (and also for a generic timeindependent perturbation of a quasi-convex integrable Hamiltonian with n = 3) based on his variational techniques. Mather never gave a complete proof of the announced results, but his work and unpublished preprints played a fundamental role in the subsequent developments. First, Bernard, Kaloshin and Zhang in [START_REF] Bernard | Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders[END_REF] proved a weaker form of Arnold conjecture, still with the convexity requirement but for an arbitrary number of degrees of freedom. Then, Kaloshin and Zhang in [START_REF] Kaloshin | A strong form of Arnold diffusion for two and a half degrees of freedom[END_REF] proved the strong form of Arnold conjecture, for n = 2, h convex and f time-periodic. A different approach to this problem was proposed by Cheng ([Che13]) and announced by Marco ([Mar12a], [Mar12b]).

Role of normally hyperbolic invariant cylinders

The central and common point in all these works, which was not present in the work of Mather, is the use of normally hyperbolic invariant manifolds as a "skeleton" for the unstable orbits. Construction of such cylinders under generic hypothesis relies on resonant normal forms and the theory of normally hyperbolic manifolds, and they were already discussed in [START_REF] Kaloshin | Almost dense orbit on energy surface[END_REF] and [START_REF]Large normally hyperbolic cylinders in a priori stable Hamiltonian systems[END_REF].

On the other hand, most of these works do rely strongly on Mather's variational techniques, once the normally hyperbolic invariant manifolds have been constructed. It has to be noted that these variational techniques, and to a lesser extent, the existence of normally hyperbolic cylinders, use in an essential way the convexity assumption, so that none of these works apply to non-convex integrable Hamiltonians.

Non-convex systems

It can be said that a typical non-degenerate integrable system (in the sense of Kolmogorov, Arnold or Nekhoroshev) is non-convex nor quasi-convex, but for these systems, essentially nothing is known: for the simplest integrable Hamiltonians h which are non-convex nor quasi-convex but steep and non-degenerate (in the sense of Kolmogorov for n ≥ 2 or Arnold for n ≥ 3), it is not even known how to construct a single f such that H = h+ εf has unstable orbits. This is a bit paradoxical from the point of view of Nekhoroshev estimates, as the time of diffusion for perturbations of steep non-convex integrable Hamiltonians should be smaller and hence diffusion should be easier to observe.

Another evidence of the difficulty connected to the lack of convexity is the problem of existence of periodic orbits (not to mention, of course, the problem of extending Aubry-Mather theory). Recall that Bernstein and Katok proved, in [START_REF] Bernstein | Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians[END_REF], that given a periodic tori (filled with periodic orbits of common period) of a convex integrable Hamiltonian system with n degrees of freedom, at least n periodic orbits persist after a small perturbation, no matter how large is the period (that is, the threshold of the perturbation is independent of the period). Moreover, these periodic orbits are uniformly (with respect to the period) close to the unperturbed ones. For non-convex non-degenerate integrable Hamiltonian system, this was partly generalized in [START_REF] Chen | Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with nondegenerate Hessian, Twist mappings and their applications[END_REF]: one can still get the existence of at least n periodic orbits but without the uniform estimates (one can find in [START_REF] Chen | Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with nondegenerate Hessian, Twist mappings and their applications[END_REF] an example due to Herman which shows that these uniform estimates are indeed not possible without convexity).

Examples of non-convex non-steep Hamiltonians

Yet for some non-convex non-steep integrable Hamiltonians, the construction of examples of instability is much easier and has been known for a long time. A prototype of such an integrable Hamiltonian with two degrees of freedom, which can be found in [START_REF]An exponential estimate of the time of stability of nearly integrable Hamiltonian systems[END_REF] (but a completely analogous example, in a slightly different setting, was already considered in [START_REF] Moser | On the elimination of the irrationality condition and Birkhoff 's concept of complete stability[END_REF]), is given by h(I 1 , I 2 ) = 1 2 (I 2 1 -I 2 2 ): letting f (θ 1 , θ 2 ) = (2π) -1 sin(2π(θ 1 -θ 2 )), the system H = h + εf admits the unstable solution I(t) = (-εt, εt), θ(t) = -1 2 (εt 2 , εt 2 ). This Hamiltonian h is obviously non-convex, but it is also non-steep since the restriction of h to the lines {I 1 ± I 2 = 0} is constant so this restriction has only critical points, which are thus non-isolated. Also it is degenerate in the sense of Arnold, so diffusion can and do already occur for n = 2, even though it is non degenerate in the sense of Kolmogorov so that it admits many invariant tori (circles). Moreover, the time of diffusion in this example is the smallest possible, as it is linear with respect to the inverse of ε. Let us point out that for integrable systems with two degrees of freedom, the Arnold non-degeneracy condition is in fact equivalent to quasi-convexity which is also equivalent to steepness.

It is obvious that the above example can be generalized to the case where h is a quadratic form that has an isotropic vector with rational components (such a quadratic form is thus indefinite) for any number of degrees of freedom n, but in fact more is true. On the one hand, it was proved by Nekhoroshev in [START_REF]An exponential estimate of the time of stability of nearly integrable Hamiltonian systems II[END_REF] that one can consider an even more general class of integrable Hamiltonians. Indeed, suppose there exist an affine subspace L of R n , whose associated vector space is spanned by vectors with rational components, and a curve σ : [0, 1] → L such that the gradient of the restriction of h to L vanishes identically along σ. Then one can construct an arbitrarily small perturbation εf such that the system H = h + εf has an orbit (θ(t), I(t)) for which I(0) = σ(0) and I(τ ) = σ(1), where τ proportional to ε -1 .

On the other hand, in the case where h is a quadratic form with a rational isotropic vector, Herman had constructed examples of perturbation for which one can find a dense G δ set of initial conditions leading to orbits whose action components are unbounded (note that such a quadratic form can be non-degenerate, so at the same time most of the orbits lie on invariant tori). We refer to [START_REF] Herman | Dynamics connected with indefinite normal torsion, Twist mappings and their applications[END_REF] for such examples and many other interesting examples related to non-convex integrable Hamiltonians.

Preliminary description of the class of Hamiltonians studied in the paper

The examples above are rather specific. For instance, for the prototype of non-convex integrable Hamiltonian h(I 1 , I 2 ) = 1 2 (I 2 1 -I 2 2 ), the perturbation f does not depend on the action variables and more importantly, it depends only on a specific combination of the angular variables. The purpose of this paper is to investigate the question whether such a phenomenon remains true for a generic perturbation. We will show in Theorem 1.1 that we have diffusion for a class of non-convex non-steep Hamiltonians h with two degrees of freedom, which includes the example h(I 1 , I 2 ) = 1 2 (I 2 1 -I 2 2 ) as a particular case, and for an open and dense set of perturbations, with a time of diffusion which is linear with respect to the inverse of ε. Under stronger assumptions, we will also prove a stronger statement of diffusion in Theorem 1.2. The conditions defining this class of integrable Hamiltonians h is the existence of a segment with rational slope contained in an energy level of h such that the gradient ∇h does not vanish along this segment (these are the assumptions (A.1) and (A.2) in §1.2). For integrable Hamiltonians which are compatible with "fast" diffusion (that is, with a time of diffusion which is linear with respect to the inverse of ε) for some perturbation, we expect these conditions to be quite sharp. The set of admissible perturbations is very easily described: we only required that some "averaged" perturbation is a non-constant function. Moreover, we only require h and f to be of finite regularity.

Heuristic description of the proof

Let us now explain the idea of the proof on the specific example h(I 1 , I 2 ) = I 2 1 -I 2 2 . Up to a linear symplectic change of coordinates, we can equivalently consider h(I 1 , I 2 ) = I 1 I 2 . If the perturbation f depends only on θ 1 and is non-constant, then we are essentially back to the example described in §1.1.7. Indeed, the equations of motion in this case are

İ1 (t) = -εf ′ (θ 1 (t)), İ2 (t) = 0, θ1 (t) = I 2 (t), θ2 (t) = I 1 (t)
so that if we choose an initial condition (θ(0), I(0)) with f ′ (θ 1 (0)) = 0 (which is possible since f is non constant) and I 2 (0) = 0, then I 2 (t) = 0 and θ 1 (t) = θ 1 (0) for all time t ∈ R and so the I 1 variable drift with a speed of order ε:

I 1 (t) = I 1 (0) -tεf ′ (θ 1 (0)).
One should observe that this drift happens because the solution is locked in the resonance {I 2 = 0} for all time. Now for a general perturbation f depending on (θ, I), on a suitable domain one can average the θ 2 variables: more precisely the Hamiltonian can be conjugated to a Hamiltonian of the form h(I) + ε f (θ 1 , I) + ε 2 f ′ (θ, I), where f is obtained from f by averaging over θ 2 . This is close to the example we described before, except that f depends also on the action variables and there is a remainder of order ε 2 , so we cannot find a solution which stays locked in the resonance {I 2 = 0} for all time. However, it is still possible to find a solution which stays ε-close to the resonance {I 2 = 0} for a time of order ε -1 and this is sufficient to prove that the I 1 variable can drift independently of ε, provided of course that the function θ 1 → f (θ 1 , I) is non-constant for a fixed value of I (this is satisfied for an open and dense set of perturbations).

The main difficulty is that usually, the domain on which one can conjugated the Hamiltonian to the special form we described above (which is called a resonant normal form with a remainder) is ε-dependent and in the space of action, its diameter goes to zero as ε goes to zero. We will actually prove (in Proposition 2.3) that in our situation, we can construct such a conjugacy on a domain in the action space which contains a segment, in the I 1 direction, whose length is independent of ε, so that the existence of a drifting orbit for the normal form will actually yield the existence of a drifting orbit for the original Hamiltonian (this is Theorem 1.1). Assuming moreover that f is action-independent and constructing a more accurate normal form with a remainder of order ε 3 instead of ε 2 (as in Proposition 2.4), we will also show that the size of the drift can be as large as we want (this is Theorem 1.2).

Let us point out that the main ingredient of the proof, which is the normal form, is similar in spirit to the normal form constructed in [START_REF] Bernard | Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders[END_REF]. However, in [START_REF] Bernard | Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders[END_REF] the normal form is used for another purpose (namely to construct a normally hyperbolic invariant cylinder which is then used to locate an unstable orbit) so they only need the remainder to be of order δε, for some δ > 0 independent of ε. In our case, we need a stronger statement (a remainder of order at least ε 2 ) in order to derive the existence of an unstable orbit directly from the normal form.

Prospects

To conclude, let us note that the statement of Theorem 1.1 gives a diffusion in a weak sense, that is the action variables drift independently of ε for all ε sufficiently small, but we cannot find an orbit which connects arbitrary neighbourhoods in the space of action. Also, for the moment, it is restricted to two degrees of freedom, which is the minimal number of degrees of freedom for which instability can occur for Arnold degenerate integrable systems. The normal form we used is in fact valid for any number of degrees of freedom, but in general it appears too weak to derive the result directly from it, and therefore we expect that additional restrictions on the set of admissible perturbations has to be imposed for more degrees of freedom. We plan to come back to these issues in a subsequent work.

Main results

Geometric assumptions

Given R > 0, let B R be the closed ball of R 2 of radius R with respect to the supremum norm | . |, that is B R = {(I 1 , I 2 ) ∈ R 2 | |I 1 | ≤ R, |I 2 | ≤ R}.
Our integrable Hamiltonian h will be a function h : B R → R of class C 4 , which satisfy the following two conditions:

(A.1) There exist a vector k = (k 1 , k 2 ) ∈ Z 2 \ {0}, a constant a ∈ R and a closed segment S ⊆ L ∩ B R , where L = {(I 1 , I 2 ) ∈ R 2 | k 1 I 1 + k 2 I 2 + a = 0}, such that the restriction of h to S is constant. (A.
2) There exists a closed segment S * ⊆ S, such that for all I ∈ S * , ∇h(I) = 0. Such a segment S is sometimes called a channel of superconductivity. Note that the condition (A.1) obviously rules out convex functions, but it also rules out steep functions. Indeed, (A.1) is equivalent to the assertion that the gradient of h |L , the restriction of h to L, vanishes identically on S, therefore the function h |L has a set of critical points which contains S and hence is non-isolated. As for the condition (A.2), it is a non-degeneracy assumption, as we want to avoid that the gradient of h vanishes identically on S: note that (A.2) is satisfied if there exists I * ∈ S such that ∇h(I * ) = 0. The condition (A.1) is crucial, whereas (A.2) is somehow just technical, as we believe it can be removed in general. Following the terminology of [START_REF] Bounemoura | Optimal stability and instability for near-linear hamiltonians[END_REF], functions which do satisfy (A.1) are functions which are not rationally steep.

Regularity assumptions

Given a small parameter 0 < ε < 1, our perturbation εf will be a "generic" function εf : T 2 × B R → R which is "small" for the C r topology, for r sufficiently large. For an integer

r ≥ 2, let C r (T 2 × B R ) the space of C r function f : T 2 × B R → R, which is Banach space with respect to the norm |f | C r (T 2 ×B R ) = sup j∈N 4 , |j|≤r sup (θ,I)∈T n ×B R |∂ j f (θ, I)|
where we have used the standard multi-index notation. We extend the definition of the C rnorm for vector-valued functions

F = (f 1 , . . . , f m ) : T 2 × B R → R m , for an arbitrary integer m ≥ 1, by setting |F | C r (T 2 ×B R ,R m ) = sup 1≤i≤m |f i | C r (T 2 ×B R ) .
Let us denote by C r 1 (T 2 × B R ) the unit ball of C r (T 2 × B R ) with respect to this norm, that is

C r 1 (T 2 × B R ) = {f ∈ C r (T 2 × B R ) | |f | C r (T 2 ×B R ) ≤ 1}.
Our perturbation εf will be such that f belongs to an open and dense subset F r k of C r 1 (T 2 × B R ), depending on the vector k defined in (A.1). For a given function f ∈ C r 1 (T 2 × B R ) and a given I * in the interior of S * , we define f *

k ∈ C r 1 (T 2 ) by f * k (θ) = 1 0 f (θ + tk, I * )dt, then F r k is defined by F r k = {f ∈ C r 1 (T 2 × B R ) | ∃ I * ∈ int(S * ), ∃ θ * ∈ T 2 , ∂ θ f * k (θ * ) = 0}.
In words, F r k is the subset of functions f ∈ C r 1 (T 2 × B R ) such that, for some I * in the interior of S * , the associated function f * k is non-constant: this is obviously an open and dense subset of C r 1 (T 2 × B R ). Note that f * k is a function on T 2 , but by definition it is constant on the orbits of the linear flow of frequency k, hence it can be considered as being defined on the space of orbits (the leaf space) of this flow, which is diffeomorphic to T.

Statements

We can finally state our first main result.

Theorem 1.1. Let H = h + εf be defined on T 2 × B R , with h ∈ C 4 1 (B R ) satisfying (A.1) and (A.2) and f ∈ F 7 k . Then there exist positive constants C ≥ 1 and c, depending only on h, and positive constants ε 0 ≤ 1 and δ ≤ 1 depending also on f , such that for any 0 < ε ≤ ε 0 , the Hamiltonian system defined by H has a solution (θ(t), I(t)) such that for τ = δε -1 ,

|I(0) -I * | ≤ cε, |I(τ ) -I(0)| ≥ Cδ 2 .
Moreover, for all t ∈ [0, τ ], we have d(I(t), S * ) ≤ cε where d is the distance induced by the supremum norm.

It is a statement of diffusion for the action variables, in the sense that they have a variation along S * ⊆ S which is bounded from below independently of ε, for all ε small enough. It has to be noted that the time of diffusion τ = δε -1 is essentially optimal in the sense that for all

f ∈ C 2 (T 2 × B R ) ∩ C 1 1 (T 2 × B R )
, for all ε > 0 and for all 0 < δ ≤ 1, we have

|I(τ ) -I(0)| ≤ δ
for all solutions of H = h + εf . In particular, for the solution given by Theorem 1.1, one has the inequalities

Cδ 2 ≤ |I(τ ) -I(0)| ≤ δ.
A stronger statement of diffusion can be reached, assuming that the perturbation is independent of the action variables and that it is slightly more regular. Indeed, let us define

G r k = {f ∈ C r 1 (T 2 ) | ∃ θ * ∈ T 2 , ∂ θ fk (θ * ) = 0}
where fk (θ) =

1 0 f (θ + tk)dt.
Then we can state our second main result.

Theorem 1.2. Let H = h + εf be defined on T 2 × B R , with h ∈ C 10 1 (B R ) satisfying (A.1) and (A.2) and f ∈ G 19 k . Then, given any two points I ′ ∈ S * and I ′′ ∈ S * , there exists a positive constant c, depending only on h, and positive constants ε 0 ≤ 1 and δ ≤ 1 depending also on f and on the distance between I ′ and I ′′ , such that for any 0 < ε ≤ ε 0 , the Hamiltonian system defined by H has a solution (θ(t), I(t)) such that for τ ≤ δε -1 ,

|I(0) -I ′ | ≤ cε, |I(τ ) -I ′′ | ≤ cε.
Moreover, for all t ∈ [0, τ ], we have d(I(t), S * ) ≤ cε.

The conclusion of Theorem 1.2 is indeed stronger than the conclusion of Theorem 1.1, as we not only have a variation along S * ⊆ S which is independent of ε, but we can also connect ε-neighborhoods of any two points in S * . It should be noted that both theorems give a new obstruction to extend Nekhoroshev estimates in the non-steep case, even if one is willing to consider only a generic perturbation.

Concerning the dependence of the constants involved, the dependence on h is only through R, the vector k and the constant a that appeared in (A.1), the length of the segment S * that appeared in (A.2) and a lower bound on the norm of ∇h(I) for I ∈ S * (which is positive by (A.2)), while the dependence on f is through the absolute value of ∂ θ f * k (θ * ) (respectively ∂ θ fk (θ * )) for Theorem 1.1 (respectively for Theorem 1.2) and the distance of I * to the boundary of S * for Theorem 1.1. We refer to Theorem 2.1 and Theorem 2.2 in §2.1 for more concrete and precise statements.

Comments on the regularity assumptions

The first statement (Theorem 1.1) requires the integrable part to be C 4 and the perturbation to be C 7 , while the second statement (Theorem 1.2) requires the integrable part to be C 10 and the perturbation to be C 19 . Theses regularity assumptions are far from being optimal, and no efforts were made to improve them.

For instance, using analytic approximations as in [START_REF] Bernard | Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders[END_REF] or polynomial approximations, it is certainly possible to lower these regularities. Moreover, concerning the first statement, one can use a different method which would give the same result assuming only that the integrable part and the perturbation are of class C 3 (unfortunately, this method cannot be applied directly to prove the second statement).

Comments on the geometric assumptions

Let us now discuss some particular cases of functions h satisfying (A.1) and (A.2), and therefore for which one has diffusion for a generic perturbation. As we will explain later, we can always assume without loss of generality that a = 0 in (A.1), and upon adding an irrelevant additive constant, we can assume that the restriction of h to S is identically zero.

For a linear Hamiltonian h(I) = ω • I, it follows that (A.1) and (A.2) are satisfied if and only if ω is resonant, that is l • ω = 0 for some l ∈ Z 2 \ {0}, and ω is non-zero. On the other hand, if ω is non-resonant, it follows from [START_REF] Bounemoura | Optimal stability and instability for near-linear hamiltonians[END_REF] that the statement of Theorem 1.1 cannot be true since for all sufficiently small perturbation, one has stability for an interval of time which is strictly larger than [-τ, τ ] with τ as above. In particular, if ω is Diophantine, one has stability for an interval of time which is exponentially long with respect to ε -1 , up to an exponent depending only on the Diophantine exponent of ω. Now for a quadratic Hamiltonian h(I) = AI • I where A is a 2 by 2 symmetric matrix, (A.1) and (A.2) are satisfied if and only if there exists a vector l ∈ Z 2 \ {0} such that Al • l = 0 and Al = 0. Assuming that A is diagonal, its eigenvalues have to be of different sign, and writing h(I) = α 2 1 I 2 -α 2 2 I 2 2 , (A.1) and (A.2) are satisfied if and only if α 1 = 0, α 2 = 0 and α 2 /α 1 ∈ Q. The example described in the introduction corresponds to α 1 = α 2 = 1. On the other hand, one knows that if α 2 /α 1 is irrational, the statement of Theorem 2.1 cannot be true for any sufficiently small perturbation for the same reason as above: for instance, if α 2 /α 1 is a Diophantine number, the quadratic Hamiltonian falls into the class of Diophantine steep functions introduced in [Nie07] and it follows from results in [START_REF]Prevalence of exponential stability among nearly integrable Hamiltonian systems[END_REF] or [START_REF] Bounemoura | Generic Nekhoroshev theory without small divisors[END_REF] that such Hamiltonians are stable for an exponentially long interval of time.

Note that in these two special cases, the condition (A.2), which amounts to ω = 0 in the first case and Al = 0 in the second case, can be easily removed.

We already explained that the time of diffusion τ is in some sense optimal, regardless of the integrable Hamiltonian h. Now we believe that if we fix the time of diffusion, the condition (A.1) on the integrable Hamiltonian h is also in some sense optimal, as if h does not satisfy this assumption, one can have diffusion but with a time strictly greater than τ . This is indeed the case for linear or quadratic integrable Hamiltonians as we described above, and the general case is conjectured in [START_REF] Bounemoura | Optimal stability and instability for near-linear hamiltonians[END_REF].

2 Proof of Theorem 1.1 and Theorem 1.2

In §2.1, we will perform some preliminary transformations to reduce Theorem 1.1 and Theorem 1.2 to equivalent but more concrete statements, which are Theorem 2.1 and Theorem 2.2. Theorem 2.1 and Theorem 2.2 will be proved in §2.3, based on normal form results which are stated and proved in §2.2.

Preliminary reductions

Preliminary transformations

First we may assume that the line L in (A.1) passes through the origin, that is L = {(I 1 , I 2 ) ∈ R 2 | k 1 I 1 +k 2 I 2 = 0}: indeed, we can always find a translation of the action variables T :

R 2 → R 2 such that T sends {(I 1 , I 2 ) ∈ R 2 | k 1 I 1 + k 2 I 2 = 0} to {(I 1 , I 2 ) ∈ R 2 | k 1 I 1 + k 2 I 2 + a = 0},
and since the map Φ T (θ, I) = (θ, T I) is symplectic, the statement holds true for H if and only if it holds true for H • Φ T , up to constants depending on a.

Then we can suppose that the components of the vector k = (k 1 , k 2 ) ∈ Z 2 \ {0} are relatively prime, since changing k by k/p, where p is the greatest common divisor of k 1 and k 2 , does not change the definition of L. Hence we may assume that in fact k = e 2 = (0, 1), that is

L = {(I 1 , I 2 ) ∈ R 2 | I 2 = 0}: indeed we can always find a matrix M ∈ GL 2 (Z) such that its second row is k, hence M e 2 = k and t M -1 sends {(I 1 , I 2 ) ∈ R 2 | I 2 = 0} to {(I 1 , I 2 ) ∈ R 2 | k 1 I 1 + k 2 I 2 = 0}. The map Φ M (θ, I) = (M θ, t M -1 I) is well-defined since M T 2 = T 2 ,
and it is symplectic, so the statement holds true for H if and only if it holds true for H • Φ M , up to constants depending on k.

Note that the symplectic transformations Φ T and Φ M do change the domain B R in the space of actions, but to simplify the notations, we will assume that the latter is fixed.

Simplified assumptions

Now for all I = (I 1 , I 2 ) ∈ B R , let us write

∇h(I) = ω(I) = (ω 1 (I), ω 2 (I)) = (ω 1 (I 1 , I 2 ), ω 2 (I 1 , I 2 )) ∈ R 2 . Since L = {(I 1 , I 2 ) ∈ R 2 | I 2 = 0}, S, which by definition is a closed segment contained in L ∩ B R , is of the form S = {(I 1 , 0) ∈ R 2 | I 1 ∈ S 1 } where S 1 is a closed segment of R contained in [-R, R]. Similarly, S * = {(I 1 , 0) ∈ R 2 | I 1 ∈ S *
1 } where S * 1 is contained in the interior of S 1 . The condition (A.1) is then obviously equivalent to ∂ I 1 h(I 1 , 0) = ω 1 (I 1 , 0) = 0 for all I 1 ∈ S 1 , while the condition (A.2) is that ω 2 (I 1 , 0) = 0 for all I 1 ∈ S * 1 . Changing H to -H if necessary and reversing the time accordingly, we may assume that ω 2 (I 1 , 0) ≥ ̟ > 0 for all I 1 ∈ S * 1 . We can eventually formulate simplified conditions, that we call (B.1) and (B.2): (B.1) There exists a closed segment S 1 ⊆ [-R, R] such that for all I 1 ∈ S 1 , we have ω 1 (I 1 , 0) = 0. (B.2) There exist a closed segment S * 1 ⊆ S 1 and ̟ > 0 such that for all I 1 ∈ S * 1 , we have ω 2 (I 1 , 0) ≥ ̟.

Then the definition of F r e 2 also simplifies: one easily check that for f ∈ C r 1 (T 2 × B R ), we have, for some

I * = (I * 1 , 0) in the interior of S * , f * e 2 ∈ C r 1 (T) where f * e 2 (θ 1 ) = T f (θ 1 , θ 2 , I * )dθ 2
so that f ∈ F r e 2 if and only if there exist

I * = (I * 1 , 0) in the interior of S * and θ * 1 ∈ T for which ∂ θ 1 f * e 2 (θ * 1 ) = 0.
The definition of G r e 2 is analogous. For simplicity, we write f * e 2 = f * and F r e 2 = F r , fe 2 = f and G r e 2 = G r , and for f ∈ F r (respectively f ∈ G r ), we denote by λ a lower bound on the absolute value of ∂ θ 1 f * (θ * 1 ) (respectively a lower bound on the absolute value of ∂ θ 1 f (θ * 1 )). For Theorem 1.1, we denote by δ * the distance of I * 1 to the boundary of S * 1 , and for Theorem 1.2, we denote by ρ the distance between I ′ 1 ∈ S * 1 and I ′′ 1 ∈ S * 1 , where I ′ = (I ′ 1 , 0) and I ′′ = (I ′′ 1 , 0).

Simplified statements

From the previous discussion, it follows that Theorem 1.1 and Theorem 1.2 are implied by the following statements.

Theorem 2.1.

Let H = h + εf be defined on T 2 × B R , with h ∈ C 4 1 (B R ) satisfying (B.1
) and (B.2) and f ∈ F 7 . Then there exist positive constants C ≥ 1 and c depending only on R, the length of S * 1 and ̟, and a positive constant ε 0 ≤ 1 depending also on λ, such that for any 0 < ε ≤ ε 0 , if we set δ = min{λ(4C) -1 , δ * }, the Hamiltonian system defined by H has a solution (θ(t), I(t)) such that for τ = δε -1 ,

|I 1 (0) -I * 1 | ≤ cε, |I 1 (τ ) -I 1 (0)| ≥ Cδ 2 .
Moreover, for all t ∈ [0, τ ], we have |I 2 (t)| ≤ cε and d(I 1 (t), S * 1 ) ≤ cε.

Theorem 2.2. Let H = h + εf be defined on T 2 × B R , with h ∈ C 10 1 (B R ) satisfying (B.1) and (B.2) and f ∈ F 19 . Then there exists a positive constant c, depending only on R, the length of S * 1 , and ̟, and a positive constant ε 0 ≤ 1 depending also on λ and ρ, such that for any 0 < ε ≤ ε 0 , if we set δ = 2ρλ -1 , the Hamiltonian system defined by H has a solution (θ(t), I(t)) such that for τ ≤ δε -1 ,

|I 1 (0) -I ′ 1 | ≤ cε, |I 1 (τ ) -I ′′ 1 | ≤ cε.
Moreover, for all t ∈ [0, τ ], we have |I 2 (t)| ≤ cε and d(I 1 (t), S * 1 ) ≤ cε.

Normal forms

Domain of the normal forms

The main ingredient of the proofs of Theorem 2.1 and Theorem 2.2 will be normal forms on a domain which, in the space of action, is centred around S * . In particular, in the direction given by the first action variables I 1 , it contains the segment S * 1 whose length is independent of ε. For a constant κ > 0 to be determined later (in the proof of Proposition 2.3), let us consider the κε-neighbourhood of S * in R 2

S * (κε) = {I ∈ R 2 | d(I, S * ) ≤ κε} = {(I 1 , I 2 ) ∈ R 2 | d(I 1 , S * 1 ) ≤ κε, |I 2 | ≤ κε}.
We also define the domain D * (κε) = T 2 × S * (κε). Let r be an integer (we will choose r = 7 for Theorem 2.1 and r = 19 for Theorem 2.2, but for convenience we consider it as a free parameter for the moment).

In the sequel, to avoid cumbersome notations, when convenient we will use a dot • in replacement of any constant depending only on r, R, the length of S * 1 and ̟, that is for any two quantities u and v, an expression u <• v means that there exists a constant c depending only on r, R, the length of S * 1 and ̟ such that u ≤ cv. Similarly, we will use the notation u =• v. For instance, we will have κ =• 1.

To simplify the exposition further, for any integer j ≤ r we will simply denote by | . | j the C j norm of a function or a vector-valued function, without referring to its domain of definition nor to the domain where it takes values.

One-step normal form

Proposition 2.3. Let H = h + εf be defined on T 2 × B R , l ≥ 1 an integer, and assume that h ∈ C l+3 1 (B R ) satisfies (B.1) and (B.2) and f ∈ C r 1 (T 2 × B R ), for r ≥ 2(l + 1) + 3. Assume that ε <• 1. Then there exists a symplectic embedding

Φ : D * (κε/2) → D * (κε) of class C l+1 such that H • Φ = h + ε f + ε 2 f ′ , f (θ 1 , I) = T f (θ 1 , θ 2 , I)dθ 2
and we have the following estimates

|Φ -Id| 0 ≤ κε/2, |f ′ | l <• 1.
The proof of this proposition uses some elementary estimates which are recalled in the Appendix A.

Proof of Proposition 2.3. First of all, since ε <• 1, we can assume that S * (κε) is included in B R . For a function χ : D * (κε) → R of class C l+2 to be chosen below, the transformation Φ in the statement will be obtained as the time-one map of the Hamiltonian flow generated by εχ. Let X εχ be the Hamiltonian vector field generated by εχ, and X t εχ the time-t map. Assuming that X t εχ is well-defined on D * (κε/2) for |t| ≤ 1, let Φ = X 1 εχ . Using the relation

d dt G • X t εχ = ε{G, χ} • X t εχ
for an arbitrary function G, and writing

H • Φ = h • Φ + εf • Φ
we can apply Taylor's formula to the right-hand side of the above equality, at order two for the first term and at order one for the second term, to get

H • Φ = h + ε{h, χ} + ε 2 1 0 (1 -t){{h, χ}, χ} • X t εχ dt + εf + ε 2 1 0 {f, χ} • X t εχ dt = h + ε({h, χ} + f ) + ε 2 1 0 {(1 -t){h, χ} + f, χ} • X t εχ dt = h + ε f + ε({h, χ} + f -f ) + ε 2 1 0 {(1 -t){h, χ} + f, χ} • X t εχ dt (1)
where f is the function defined in the statement. It would be natural to choose χ to solve the equation {h, χ} + g = 0 where g = f -f , which can be written again as {χ, h} = g, but we will only solve this equation approximatively.

We expand f in Fourier series with respect to the variables θ:

f (θ, I) = k∈Z 2 f k (I)e 2πik•θ = (k 1 ,k 2 )∈Z 2 f (k 1 ,k 2 ) (I)e 2πi(k 1 θ 1 +k 2 θ 2 )
and for a parameter K ≥ 1 to be chosen below, we write

f (θ, I) = f K (θ, I) + f K (θ, I) = k∈Z 2 , |k|≤K f k (I)e 2πik•θ + k∈Z 2 , |k|>K f k (I)e 2πik•θ .
Instead of solving the equation {χ, h} = g with g = f -f , we will actually choose χ to solve the equation {χ, h} = g K where g K = f K -fK . Observe that

f (θ 1 , I) = k 1 ∈Z f (k 1 ,0) (I)e 2πik 1 θ 1 and hence fK (θ 1 , I) = k 1 ∈Z, |k 1 |≤K f (k 1 ,0) (I)e 2πik 1 θ 1 .
It is then easy to check that the solution of {χ, h} = g K , which can be written again as

ω(I) • ∂ θ χ(θ, I) = g K (θ, I), (θ, I) ∈ D * (κε) (2)
is given by χ(θ, I) = k∈Z 2 , |k|≤K χ k (I)e 2πik•θ where

χ k (I) = (2πik • ω(I)) -1 f k (I), |k| ≤ K, k 2 = 0 0, |k| ≤ K, k 2 = 0. ( 3
) Since h is C l+3 , χ is C l+2 provided we can prove that k • ω(I)
|k • ω( Ĩ)| = |k • ω( Ĩ1 , 0)| = |k 1 ω 1 ( Ĩ1 , 0) + k 2 ω 2 ( Ĩ1 , 0)| = |k 2 |ω 2 ( Ĩ1 , 0) ≥ ̟.
It follows that for k ∈ Z 2 such that k 2 = 0 and |k| ≤ K, and for I ∈ S * (κε), we have

|k • ω(I)| ≥ |k • ω( Ĩ)| -|k||ω(I) -ω( Ĩ)| ≥ ̟ -Kκε ≥ ̟/2 (4) provided we define K = ̟(2κε) -1 =• ε -1 . Now f ∈ C r 1 (T 2 × B R )
, so an integration by parts gives that |f k | j <• |k| j-r |f | r <• |k| j-r for any j ≤ r, and therefore using (3), (4) and Leibniz formula (inequality (19) of Appendix A), we obtain |χ k | l+1 <• |k| l+1-r . Since r ≥ 2(l + 1) + 3, we can therefore bound the C l+1 norm of χ independently of K as

|χ| l+1 <• k∈Z 2 , |k|≤K |χ k | l+1 |k| l+1 <• k∈Z 2 , |k|≤K |k| l+1-r |k| l+1 <• k∈Z 2 |k| -3 <• 1 and so |εχ| l+1 ≤ γε, γ =• 1. (5)
It is easy to see that γ is independent of κ, so we now choose κ = 2γ =• 1, and as ε <• 1, we can apply Lemma A.1 of Appendix A: for all |t| ≤ 1, X t εχ : D * (κε/2) → D * (κε) is a well-defined symplectic embedding of class C l+1 with the estimates

|X t εχ -Id| 0 ≤ κε/2, |X t εχ | l <• 1. (6)
In particular, the first estimate of (6) gives

|Φ -Id| 0 <• ε.
Now from the equalities (1) and (2) we can write

H • Φ = h + ε f + εf K -ε f K + ε 2 1 0 {(t -1)g K + f, χ} • X t εχ dt
so that, if we set

f ′ = ε -1 f K -ε -1 f K + 1 0 {(t -1)g K + f, χ} • X t εχ dt = ε -1 f K -ε -1 f K + R, then H • Φ = h + ε f + ε 2 f ′ . It remains to estimate f ′ . Using the fact that |f k | j <• |k| j-r |f | r <• |k| j-r for any j ≤ r, one easily obtain ε -1 |f K | l <• ε -1 K l+2-r <• 1 (7) by definition of K. Similarly ε -1 | f K | l <• 1. (8)
Then we have

|R| l <• |{(t -1)g K + f, χ}| l |X t εχ | l l <• |{(t -1)g K + f, χ}| l <• |(t -1)g K + f | l+1 |χ| l+1 <• 1 (9)
where we have used Faa di Bruno formula (inequality (21) of Appendix A), the last part of (6), the inequality (20) of Appendix A and the fact that

|g K | l+1 <• 1, |f | l+1 <• 1 and |χ| l+1 <• 1.
The estimates (7), (8) and (9) implies that

|f ′ | l <• 1
which concludes the proof.

Two-steps normal form

Proposition 2.4. Let H = h + εf be defined on T 2 × B R , l ≥ 1 and l ′ ≥ 1 integers, and assume that h ∈ C l+3 1 (B R ) satisfies (B.1) and (B.2) and f ∈ C r 1 (T 2 × B R ), for r ≥ 2(l + 1) + 3 and l ≥ 2(l ′ + 1) + 3. Assume that ε <• 1. Then there exists a symplectic embedding

Φ : D * (κε/4) → D * (κε) of class C l ′ +1 such that H • Φ = h + ε f + ε 2 f ′ + ε 3 f ′′ , f (θ 1 , I) = T f (θ 1 , θ 2 , I)dθ 2 , f ′ (θ 1 , I) = T f ′ (θ 1 , θ 2 , I)dθ 2
and we have the following estimates

|Φ -Id| 0 ≤ 3κε/4, |f ′ | l <• 1, |f ′′ | l ′ <• 1.
The proof of Proposition 2.4 consists essentially of applying twice Proposition 2.3; in particular the estimates are analogous so they will not be repeated below.

Proof of Proposition 2.4. First of all, the assumptions allow us to apply Proposition 2.3: there exists a symplectic embedding

Φ 1 : D * (κε/2) → D * (κε) of class C l+1 such that H 1 = H • Φ 1 = h + ε f + ε 2 f ′ , f (θ 1 , I) = T f (θ 1 , θ 2 , I)dθ 2
and we have the following estimates

|Φ 1 -Id| 0 ≤ κε/2, |f ′ | l <• 1.

Now consider the Hamiltonian H

1 = h + ε f + ε 2 f ′ defined on D * (κε/2) and of class C l+1 .
The transformation Φ in the statement will be obtained as a composition Φ = Φ 1 • Φ 2 , where Φ 2 will be the time-one map of the Hamiltonian flow generated by ε 2 χ, for some function χ : D * (κε/2) → R of class C l ′ +2 to be determined. As before, we write

H 1 • Φ 2 = h • Φ 2 + (ε f + ε 2 f ′ ) • Φ 2
and by a Taylor expansion we have

H 1 • Φ 2 = h + ε 2 {h, χ} + ε 4 1 0 (1 -t){{h, χ}, χ} • X t εχ dt + ε f + ε 2 f ′ + ε 3 1 0 { f + εf ′ , χ} • X t εχ dt = h + ε f + ε 2 f ′ + ε 2 ({h, χ} + f ′ -f ′ ) + ε 3 R.
As before also, we will choose χ to solve the equation {χ, h} = g ′ K where g

K = f ′ K -f ′ K and K =• ε -1 . Note that H 1 is of class C l+1 ,
but we have a bound only on the C l norm of f ′ , and as l ≥ 2(l ′ + 1) + 3, it can be proved that χ is C l ′ +2 with

|ε 2 χ| l ′ +1 <• ε 2

If we set

f ′′ = ε -1 f ′K -ε -1 f ′K + R,
then it can be proved that |f ′′ | l ′ <• 1 and we have

H 1 • Φ 2 = h + ε f + ε 2 f ′ + ε 3 f ′′ hence H • Φ = h + ε f + ε 2 f ′ + ε 3 f ′′ .
This concludes the proof.

2.3 Proof of Theorem 2.1 and Theorem 2.2

Proof of Theorem 2.1

The proof of Theorem 2.1 is now a consequence of the normal form Proposition 2.3. Since the latter is defined on a domain which contains the segment S * , whose length is independent of ε, it will be possible to prove the statement of Theorem 2.1 for the normal form H • Φ by analyzing directly the equation of motions, and using the fact that Φ is ε-close to the identity, we will prove that the statement remains true for H. Moreover, if we assume that ε ≤ λ(8C ′ ) -1 and ε ≤ λ 1/2 (8C ′′ ) -1/2 , then Summing up, if we define ε 0 = min{ε ′′ 0 , κ(4C ′ δ) -1 , λ(4Cδ) -1 , λ(8C ′ ) -1 , λ 1/2 (8C ′′ ) -1/2 } and c = κ, then for ε ≤ ε 0 , δ = 2ρλ -1 and τ ≤ δε -1 , the Hamiltonian H has a solution (θ(t), I(t)) for which |I 1 (0) -I ′ 1 | ≤ cε, |I 1 (τ ) -I ′′ 1 | ≤ cε. Moreover, for all t ∈ [0, τ ],

|ε 2 ∂ θ1 f ′ ( θ(
|I 2 (t)| ≤ cε, d(I 1 (t), S * 1 ) ≤ cε. This was the statement to prove.

  is non-zero, for |k| ≤ K such that k 2 = 0 and I ∈ S * (κε). By definition, given I ∈ S * (κε) we can find Ĩ ∈ S * such that |I -Ĩ| ≤ κε and therefore |ω(I) -ω( Ĩ)| ≤ κε since h ∈ C l+3 1 (B R ). For any k ∈ Z 2 such that k 2 = 0, by (B.1) we have ω 1 ( Ĩ) = ω 1 ( Ĩ1 , 0) = 0 (since Ĩ = ( Ĩ1 , 0) ∈ S * ⊆ S). Moreover by (B.2) we have ω 2 ( Ĩ1 , 0) ≥ ̟ (since Ĩ1 ∈ S * 1 ), hence

|Φ 2 -

 2 Id| 0 <• ε 2 ≤ κε/4 since ε <• 1. Therefore Φ 2 is well-defined on D * (κε/4) and of class C l ′ +1 , and so Φ : D * (κε/4) → D * (κε) is of class C l ′ +1 and satisfies|Φ -Id| 0 ≤ |Φ 1 -Id| 0 + |Φ 2 -Id| 0 ≤ 3κε/4.

Proof of Theorem 2. 1 .

 1 Recall that we are considering H = h + εf defined on T 2 × B R , with h ∈ C 4 1 (B R ) satisfying (B.1) and (B.2) and f ∈ F 7 , so we can apply Proposition 2.3 with l = 1: there exist positive constants ε ′ 0 and C ′ depending only on R, the length of S * 1 and ̟ such that if ε ≤ ε ′ 0 , there exists a symplectic embeddingΦ : D * (κε/2) → D * (κε) of class C 2 such that H • Φ = h + ε f + ε 2 f ′ , f (θ 1 , I) = T f (θ 1 , θ 2 , I)dθ 2and we have the following estimates|Φ -Id| 0 ≤ κε/2, |f ′ | 1 ≤ C ′(10)where S * (κε) ⊆ B R and D * (κε) ⊆ T 2 × B R have been defined in §2.2.Let us consider the Hamiltonian H = H • Φ defined on D * (κε/2), and we shall write Φ( θ, Ĩ) = (θ, I). Since f ∈ F 7 , there existI * = (I * 1 , 0) in the interior of S * and θ * 1 ∈ T such that |∂ θ1 f * (θ * 1 )| = |∂ θ1 f (θ * 1 , I * )| ≥ λ. (11) Since f ∈ C 19 1 (T × B R ),we obtain from the last estimate that |∂ θ1 f ( θ1 (t)) -∂ θ1 f (θ * 1 )| ≤ Cεδ, |t| ≤ τ. (17) Now recall that since f ∈ G 19 , we have |ε∂ θ1 f (θ * 1 )| ≥ ελ and assuming that ε ≤ λ(4Cδ) -1 , we have Cεδ ≤ λ/4 so that the last estimate together with (17) implies that |ε∂ θ1 f ( θ1 (t))| ≥ 3ελ/4, 0 ≤ t ≤ τ.

  and hence| Ĩ1 (τ ) -Ĩ1 (0)| = Ĩ1 (τ ) -Ĩ1 (0) ≥ τ ελ/2.We have imposed no restriction on the choice of δ, so we eventually choose δ = 2ρλ -1 . If τ = τ , then by definition of τ we have Ĩ1 (τ ) -I ′ = ρ, so Ĩ(τ ) = I ′′ and we have τ ≤ τ ′ = δε -1 . If τ = τ ′ = δε -1 , then from the last estimate and the choice of δ we obtain Ĩ1 (τ ) -Ĩ1 (0) ≥ δλ/2 = ρ which implies that Ĩ1 (τ ) -Ĩ1 (0) = ρ and τ = τ , and hence Ĩ(τ ) = I ′′ . Therefore, in any cases, τ is finite, and we have Ĩ(τ ) = I ′′ for τ ≤ δε -1 . Coming back to the original Hamiltonian, Φ( θ(t), Ĩ(t)) = (θ(t), I(t)) is a solution of the Hamiltonian H, and using the estimate on Φ, we have| Ĩ1 (t) -I 1 (t)| ≤ 3κε/4, | Ĩ2 (t) -I 2 (t)| ≤ 3κε/4as long as Ĩ(t) ∈ S * (κε/2), so in particular | Ĩ1 (0) -I 1 (0)| ≤ 3κε/4, | Ĩ1 (τ ) -I 1 (τ )| ≤ 3κε/4.

  t), Ĩ(t))| ≤ C ′ ε 2 ≤ ελ/8, |t| ≤ τ,and |ε 3 ∂ θ1 f ′′ ( θ(t), Ĩ(t))| ≤ C ′′ ε 3 ≤ ελ/8, |t| ≤ τ which gives |ε∂ θ1 f ( θ1 (t))+ε 2 ∂ θ1 f ′ ( θ(t), Ĩ (t))+ε 3 ∂ θ1 f ′′ ( θ(t), Ĩ(t))| ≥ 3ελ/4-ελ/8-ελ/8 = ελ/2, |t| ≤ τ.

From the equations of motion of H, this implies

d dt Ĩ1 (t) ≥ ελ/2, |t| ≤ τ,

Note that necessarily λ ≤ 1 since f ∈ C 7 1 (T 2 × B R ), and recall that δ * is the distance of I * to the boundary of S * . Since H is C 2 , we can consider a solution ( θ(t), Ĩ(t)) of the system defined by H with an initial condition ( θ(0), Ĩ (0)) such that Ĩ(0) = I * , θ1 (0) = θ * 1 and θ2 (0) ∈ T is arbitrary: we have the equations

since f is independent of the second angular variables. For a positive constant δ to be chosen later in terms of λ and δ * , we let τ = δε -1 . From the second equation of ( 12) and the first estimate of (10), we get

Therefore, from the third equation of ( 12), the second estimate of (10) and the fact that f ∈ C 7 1 (T × B R ), we have

Moreover, recall that | Ĩ2 (t) -Ĩ2 (0)| ≤ C ′ εδ ≤ Cδ for |t| ≤ τ by the definition of C and since ε ≤ 1, and from the first equation of (12), the first estimate of (10) and the fact that f ∈ C 7 1 (T × B R ), we also have

by the definition of C and since ε ≤ 1 and therefore

Using the fact that f ∈ C 7 1 (T × B R ), from ( 13) and ( 14) we obtain

We eventually choose δ = min{λ(4C) -1 , δ * }, and hence τ = δε -1 ≤ λ(4C) -1 ε -1 . We have to make sure that δ ≤ 1 and δ ≤ min{δ * , κ(2C ′ ) -1 }. The first requirement is obviously satisfied since C ≥ 1 and λ ≤ 1, and hence δ ≤ λ(4C) -1 ≤ 1. For the second one, which reduces to 11), (15) and the definition of δ, we have for all |t| ≤ τ ,

Moreover, if we assume that ε ≤ (4C ′ ) -1 λ, then from the second estimate of (10), we have

and this gives, as before,

Now from the first equation of ( 12), we obtain

Coming back to the original Hamiltonian, Φ( θ(t), Ĩ(t)) = (θ(t), I(t)) is a solution of the Hamiltonian H, and from the first estimate of (10), we have

as long as Ĩ(t) ∈ S * (κε/2), so in particular

Summing up, if we define

Moreover, for all t ∈ [0, τ ],

This was the statement to prove.

Proof of Theorem 2.2

The proof of Theorem 2.2 is based on the normal form Proposition 2.4, and is similar to the proof of Theorem 1.1 so we will not repeat several details. The only difference is that the perturbation f (and hence its average f ) is action independent so that the action dependence in the normal form is of order ε 2 , and moreover the dependence on θ 2 in the normal form is of order ε 3 . These facts will be used to control the solution on a longer time τ , and this will eventually give a larger drift of the action variable I 1 .

Proof of Theorem 2.2. Recall that we are considering H = h + εf defined on T 2 × B R , with h ∈ C 10 1 (B R ) satisfying (B.1) and (B.2) and f ∈ G 19 , so we can apply Proposition 2.3 with l = 7 and l ′ = 1: there exist positive constants ε ′′ 0 , C ′ and C ′′ depending only on R, the length of S * 1 and ̟ such that if ε ≤ ε ′ 0 , there exists a symplectic embedding

and we have the following estimates

Let I ′ ∈ S * and I ′′ ∈ S * , without loss of generality we may assume that

As before, we consider a solution ( θ(t), Ĩ(t)) of the system defined by H = H • Φ with an initial condition ( θ(0), Ĩ (0)) such that Ĩ(0) = I ′ , θ1 (0) = θ * 1 and θ2 (0) ∈ T is arbitrary, and for a positive constant δ to be chosen later in terms of λ and ρ, we let τ ′ = δε -1 . Moreover, we define

and we let τ = min{τ ′ , τ } ∈ R + . Reversing time if necessary, we may assume that I ′ 1 ≤ Ĩ1 (τ ). Note that in the normal form, only f ′′ depends on θ 2 , so from the equations of motion of H and the estimate on f ′′ ,

and hence, using the fact that f is action-independent,

A Technical estimates

Let S * be a bounded domain in R 2 , and for 0 < ε < 1 and a positive constant κ, consider the domains S * (κε) = {I ∈ R 2 | d(I, S * ) ≤ κε} and D(κε) = T 2 × S * (κε).

Let us begin by recalling some elementary estimates. First if f ∈ C r (D(κε)) for r ≥ 2, then for j ∈ N 4 , |j| ≤ r, ∂ l f ∈ C r-|j| (D(κε)) and obviously

In particular, this implies that if f ∈ C r (D(κε)), then its Hamiltonian vector field X f is of class C r-1 and

Then, given two functions f, g ∈ C r (D(κε)), the product f g belongs to C r (D(κε)) and by the Leibniz formula

for some constant depending only on r. By ( 18) and ( 19), the Poisson Bracket {f, g} belongs to C r-1 (D(κε)) and

for another constant c(r) depending only on r.

Given any two vector-valued functions F and G of class C r , defined on appropriate domains in R m with values in R m , such that the composition F • G makes sense, from Faa di Bruno's formula (see for instance [START_REF] Abraham | Transversal mappings and flows[END_REF]) it is easy to see that

for a constant c(m, r) depending only on m and r. Faa di Bruno's formula and classical results on the existence and regularity of solutions of differential equations can be used to prove the following lemma. Note that the constants C depend only on c, κ and on the diameter of S * (κε), but since ε < 1, the latter is bounded by d + 2κ where d is the diameter of S * .

The proof of the above lemma is a simple adaptation of Lemma 3.15 in [START_REF] Delshams | Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems[END_REF], see also Lemma A.1 in [START_REF]Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians[END_REF].