Geometry and Dynamics of Configuration Spaces Mickaël Kourganoff

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Geometry and Dynamics of Configuration Spaces Mickaël Kourganoff Geometry and dynamics of configuration spaces Mickaël Kourganoff To cite this version: Mickaël Kourganoff. Geometry and dynamics of configuration spaces. Differential Geometry [math.DG]. Ecole normale supérieure de lyon - ENS LYON, 2015. English. NNT : 2015ENSL1049. tel-01250817 HAL Id: tel-01250817 https://tel.archives-ouvertes.fr/tel-01250817 Submitted on 5 Jan 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. These` en vue de l’obtention du grade de Docteur de l’Universit´ede Lyon, d´elivr´epar l’Ecole´ Normale Sup´erieurede Lyon Discipline : Math´ematiques pr´esent´eeet soutenue publiquement le 4 d´ecembre 2015 par Micka¨el Kourganoff G´eom´etrieet dynamique des espaces de configuration Directeur de th`ese: Abdelghani Zeghib Devant la commission d’examen form´eede : Viviane Baladi (Universit´ePierre et Marie Curie, Paris), examinatrice Thierry Barbot (Universit´ed’Avignon), rapporteur G´erard Besson (Universit´ede Grenoble), examinateur Etienne´ Ghys( Ecole´ Normale Sup´erieure de Lyon), examinateur Boris Hasselblatt (Tufts University), rapporteur Mark Pollicott (University of Warwick), rapporteur BrunoS evennec´ ( Ecole´ Normale Sup´erieure de Lyon), examinateur Abdelghani Zeghib( Ecole´ Normale Sup´erieure de Lyon), directeur Unit´ede Math´ematiques Pures et Appliqu´ees– Ecole´ doctorale InfoMaths 2 Remerciements Mes premiers remerciements vont `aGhani, qui m’a fait d´ecouvrir un sujet passionnant, riche et original, et m’a laiss´eune grande libert´e,tout en sachant s’impliquer dans le d´etaild`esque n´ecessaire. Merci aussi pour toutes ces heures pass´eesdans ton bureau `a m’expliquer avec patience et enthousiasme toutes ces math´ematiques magnifiques. Je suis tr`eshonor´eque Thierry Barbot, Boris Hasselblatt et Mark Pollicott aient accept´ed’ˆetrerapporteurs. Merci pour leur travail minutieux de relecture et leurs remarques qui ont contribu´e`aam´eliorer ce manuscrit. Merci ´egalement `aViviane Baladi, G´erardBesson, Etienne´ Ghys et Bruno S´evennec d’avoir accept´ede faire partie du jury. Merci `atous les membres de l’UMPA, et en particulier aux g´eom`etres, toujours prˆets`a partager ce qu’ils savent, et qui ont toujours accept´emes invitations `aparler au S´eminaire Introductif Pr´eparatoire Intelligible et Compr´ehensible (SIPIC) : Etienne,´ Ghani, Bruno, Jean-Claude, Damien, Olivier, Alexei, Emmanuel, Marco, Romain. Merci aussi `a tous les th´esards, post-docs ou AGPR grˆace`aqui l’ambiance est si agr´eable: Agathe, Alessandro, Alvaro, Anne, Arthur, Emeric,´ Fangzhou, Fran¸cois,L´eo,Lo¨ıc, Marie, Marielle, Maxime, Michele, Olga, R´emi,Romain, Samuel, S´ebastien, Valentin, Vincent, et tous les autres. Merci `aMagalie et Virginia dont la gentillesse et l’efficacit´ene sont plus `aprouver. L’ENS Lyon est ´egalement un cadre privil´egi´epour d´ebuter en tant qu’enseignant : merci `atoute l’´equipe enseignante, aux ´el`eves, et `aceux qui ont pr´epar´eles TD avec moi et avec qui j’ai le plaisir d’´echanger des id´eeset des exercices (Alexandre et Alexandre, Daniel, Mohamed et Sylvain). Mais je dois aussi beaucoup aux membres de l’Institut Fourier `aGrenoble, o`uj’ai pass´ebeaucoup de temps durant la seconde moiti´ede ma th`ese.Merci `aG´erardBesson pour son accueil chaleureux, `aPierre Dehornoy pour avoir relu et aid´e`aam´eliorer un passage de ma th`ese,et `aBenoˆıt Kloeckner et Fr´ed´ericFaure pour des discussions fructueuses. Merci `aK´evinet Simon qui ont partag´emon bureau `aGrenoble et l’ont rendu vivant, mais aussi `aAlejandro, Bruno, Cl´ement, Guillaume, Simon, Thibaut et aux autres th´esards grenoblois. Merci `aDaniel et `ama m`ered’avoir corrig´eun certain nombre de maladresses de mon anglais dans cette th`ese. Merci `aJos Leys, qui contribue b´en´evolement depuis des ann´ees`ala recherche math´ematique, en cr´eant des images et des vid´eosde grande qualit´equi rendent les math´ematiques plus concr`eteset accessibles. Ce fut un grand plaisir de travailler avec lui : je lui dois toutes les images du chapitre 8, ainsi qu’une vid´eoqui illustre le r´esultat principal de ce chapitre. Merci `aPierre Joly pour m’avoir transmis sa passion des maths il y a bien longtemps. Merci `aceux avec qui j’ai eu la chance de faire de la belle musique pour me changer les id´eespendant ces ann´eesde th`ese: les ´el`eves et les profs des classes de chant et de 3 4 direction des conservatoires de Villeurbanne et Grenoble, les chanteurs de l’UMPA et du LIP avec qui j’ai d´ecouvert l’improvisation dans le style de la renaissance, les chanteurs d’Emelth´ee,les´ membres des chœurs que j’ai eu grand plaisir `adiriger `aSaint-Cyr au Mont d’Or et `aMeylan, tous les musiciens intrigu´espar le m´etier de chercheur en math´ematiques. Merci aux membres du Raton-Laveur ou assimil´es(Aisling, Aurore, Benjamin, Benoˆıt, Camille,Elodie,´ Guillaume, Guillaume, H´el`ene,Ir`ene,Jonas, Marthe, Laetitia, Mika¨el,Nathana¨el,Oph´elia,Pierre, Quentin, R´emi,Timoth´eeet les autres) pour tous ces mails inutiles dont je ne pourrais pas me passer, pour ces randonn´ees,laser games, parties de Sporz, LANs de tetrinet ou concours de xjump. Merci `aMartin, Margaret, Guillaume, Samuel, Sol`ene,Cl´emence,et les autres qui ont ´egay´emes week-ends de retour `aParis. Merci `atous les membres de ma famille pour leur soutien, et pour m’avoir gentiment ´ecout´eleur expliquer en quoi le m´ecanisme de Peaucellier ´etaitune invention g´eniale. Enfin, merci `aLaetitia, pour sa relecture des pages de ma th`esequi contiennent le mot “graphe”, et surtout, pour tout le reste. Contents 1 What is a linkage? 11 1.1 Fundamental examples . 12 1.2 Straight-line motion . 17 I Universality theorems for linkages in homogeneous surfaces 23 2 Introduction and generalities on universality 25 2.1 Some historical background . 25 2.2 Results . .................... 27 2.3 Ingredients of the proofs . 30 2.4 Algebraic and semi-algebraic sets . 33 2.5 Generalities on linkages . 34 2.6 Regularity . 36 2.7 Changing the input set . 36 2.8 Combining linkages . 37 2.9 Appendix: Linkages on any Riemannian manifold . 39 3 Linkages in the Minkowski plane 41 3.1 Generalities on the Minkowski plane . 41 3.2 Elementary linkages for geometric operations . 43 3.3 Elementary linkages for algebraic operations . 51 3.4 End of the proof of Theorem 2.4 . 54 4 Linkages in the hyperbolic plane 57 4.1 Generalities on the hyperbolic plane . 57 4.2 Elementary linkages for geometric operations . 58 4.3 Elementary linkages for algebraic operations . 63 4.4 End of the proof of Theorem 2.6 . 66 5 Linkages in the sphere 69 5.1 Elementary linkages for geometric operations . 69 5.2 Elementary linkages for algebraic operations . 73 5.3 End of the proof of Theorem 2.9 ford=2.................. 77 5.4 Higher dimensions . 78 5 6 CONTENTS II Anosov geodesicflows, billiards and linkages 81 6 Anosov geodesicflows and dispersing billiards 83 6.1 Introduction . 83 6.2 The cone criterion . 86 6.3 Anosov geodesicflows ................... 89 6.4 Smooth dispersing billiards . 94 7 Geodesicflows offlattened surfaces 97 7.1 Introduction . 97 7.2 Main results . 98 7.3 Proof of Theorem 7.1 . 100 7.4 Proof of Theorem 7.2 . 108 8 Dynamics of linkages 111 8.1 Introduction . 111 8.2 Proof of Theorem 8.3 . 115 III Transverse similarity structures on foliations 121 9 Transverse similarity structures on foliations 123 9.1 Some background and vocabulary . 123 9.2 Introduction . 123 9.3 Foliations with transverse similarity structures . 129 9.4 End of the proofs of the main results . 132 Introduction (fran¸cais) Cette th`eseest divis´eeen trois parties qui peuvent ˆetrelues ind´ependamment. Dans la premi`ere,on ´etudie les th´eor`emesd’universalit´epour les m´ecanismes (aussi appel´es syst`emesarticul´es) dont l’espace ambiant est une surface homog`ene.Dans la seconde, on ´etudie un lien entreflots g´eod´esiques et billards, ainsi que la dynamique de certains m´ecanismes. La troisi`emeporte sur les structures de similitude transverses sur les feuilletages, ainsi que sur le th´eor`emede d´ecomposition de De Rham. Chacune de ces parties contient une introduction propre. Un m´ecanisme est un ensemble de tiges rigides reli´ees par des liaisons pivots. Math´ematiquement, on consid`ereun m´ecanisme comme un graphe muni d’une structure suppl´ementaire : on associe une longueur `achaque arˆete(le graphe est dit m´etrique), et certains sommets sontfix´esau plan tandis que d’autres ´evoluent librement. Une r´ealisation d’un m´ecanisme dans le plan est le choix d’une position dans le plan pour chaque sommet, de sorte que les longueurs associ´eesaux arˆetescorrespondent aux distances dans le plan entre les sommets correspondants. En particulier, on autorise les arˆetes`ase croiser. Enfin, l’espace de configuration d’un syst`emearticul´eest l’ensemble de ses r´ealisations. Le premier chapitre constitue une introduction `acette notion de m´ecanisme : on y donne des ´el´ements historiques, et des exemples fondamentaux, qui sont utiles pour les chapitres suivants. La premi`erepartie, qui suit cette introduction, est constitu´eede quatre chapitres et correspond `ala pr´e-publication [Kou14] : on y ´etudie des m´ecanismes dont l’espace ambiant n’est plus le plan, mais diverses vari´et´esriemanniennes.
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