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Partially Hyperbolic Diffeomorphisms with a Compact Center Foliation with Finite Holonomy Doris Bohnet

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Partially Hyperbolic Diffeomorphisms with a Compact Center Foliation with Finite Holonomy Doris Bohnet Partially hyperbolic diffeomorphisms with a compact center foliation with finite holonomy Doris Bohnet To cite this version: Doris Bohnet. Partially hyperbolic diffeomorphisms with a compact center foliation with finite holon- omy. Dynamical Systems [math.DS]. Universität Hamburg, 2011. English. tel-00782664 HAL Id: tel-00782664 https://tel.archives-ouvertes.fr/tel-00782664 Submitted on 30 Jan 2013 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. Partially hyperbolic systems with a compact center foliation with finite holonomy Dissertation Zur Erlangung des Doktorgrades der Fakult¨at f¨ur Mathematik, Informatik und Naturwissenschaften der Universit¨at Hamburg vorgelegt im Fachbereich Mathematik von DORIS BOHNET aus Konstanz Hamburg 2011 Als Dissertation angenommen vom Fachbereich Mathematik der Universit¨at Hamburg aufgrund der Gutachten von: Dr. Roland Gunesch, Prof. Dr. Christian Bonatti Hamburg, den 12.12.2011. Prof. Dr. Ingenuin Gasser Leiter des Fachbereichs Mathematik Contents Vorwort 6 Introduction 11 1 Preliminaries 16 1.1 Compact foliations and homeomorphism groups . 16 1.2 Partiallyhyperbolicsystems . 26 1.2.1 Definitionsandbasicproperties . 26 1.2.2 Dynamical coherence for partially hyperbolic systems with a compact center foliation with finite holonomy . 30 1.2.3 Furtherremarks. 35 1.3 Dynamicsontheleafspace. 37 1.3.1 Compact center leaves with trivial holonomy . 39 1.3.2 Compact center leaves with finite holonomy . 44 1.3.3 Shadowing Lemma for compact center foliations with finiteholonomy ...................... 46 1.4 Orientabilityandcoveringspaces . 69 2 Compact center foliations with finite holonomy under restric- tions on the codimension 72 2.1 Maintheorems .......................... 72 2.2 Generallemmata ......................... 75 2.2.1 Transitivityontheleafspace . 78 2.3 Codimension-2centerfoliation. 84 2.4 Codimension-3centerfoliation . 88 2.5 Codimension-(1+k)centerFoliation . 93 2.6 Codimension-(2+2)centerFoliation . 113 Outlook 118 Bibliography 122 3 Contents Zusammenfassung 123 Lebenslauf 124 4 List of Figures 1.1 Failureofdynamicalcoherence. 32 2.1 Orientation-reversingholonomy . 77 2.2 Proofofcentraltransitivity(1) . 82 2.3 Proofofcentraltransitivity(2) . 84 2.4 Accumulationofunstableplaques . 90 5 Vorwort Die vorliegende Arbeit ist der Klassifikation partiell hyperbolischer Diffeo- morphismen gewidmet. Die Untersuchung partiell hyperbolischer Diffeomor- phismen ist ein Teil der Theorie der dynamischen Systeme. Unter einem dynamischen System versteht man ein beliebiges System, zum Beispiel ein physikalisches oder ein ¨okonomisches, das sich in Abh¨angigkeit von der Zeit ver¨andert. Wir betrachten nur dynamische Systeme mit diskreter Zeit. Dem- nach l¨asst sich ein dynamisches System durch den Raum M der Zust¨ande des Systems und durch eine Abbildung f : M → M beschreiben, die die Ver¨anderung des Systems in einer Zeiteinheit erfasst. Diese Arbeit beschr¨ankt sich auf differenzierbare dynamische Systeme, bei denen f ein Diffeomorphis- mus und M eine kompakte glatte Mannigfaltigkeit ist. Die Theorie der dynamischen Systeme untersucht das langfristige Verhal- ten eines Systems: Gibt es einen Zustand oder eine Menge von Zust¨anden, denen sich das System unabh¨angig von seinem Anfangszustand ann¨ahert? Ist das System stabil gegen¨uber kleinen St¨orungen von f oder geringf¨ugigen Anderungen¨ des Anfangszustandes? In den 1960er Jahren hat man fest- gestellt, dass strukturstabiles dynamisches Verhalten eng mit dem Begriff der Hyperbolizit¨at korreliert. In den darauffolgenden Jahren sind deshalb sogenannte hyperbolische dynamische Systeme intensiv studiert worden, und ihr Verhalten ist heute gut verstanden. In der Folge tauchten in den 1970er Jahren Ideen auf, den Begriff der Hyperbolizit¨at abzuschw¨achen, so dass eine gr¨oßere Klasse von Systemen erfasst wird, aber dennoch viele Eigenschaften hyperbolischer Systeme erhalten bleiben. In diesem Zusammenhang sei an den Begriff der partiellen Hyperbolizit¨at erinnert, eingef¨uhrt von Pugh und Shub in [PS72] und unabh¨angig davon von Brin und Pesin in [BP74], sowie an den Begriff der nichtuniformen Hyperbolizit¨at von Pesin in [Pes77], heute auch als Pesin-Theorie bekannt, und an das dominated splitting, entwickelt von Ma˜n´ein [Ma˜n84]. Der Gegenstand dieser Arbeit sind differenzierbare partiell hyperbolische Sys- teme. Ein glattes dynamisches System f : M → M wird partiell hyperbolisch 6 Vorwort genannt, falls sein Tangentialb¨undel in drei nichttriviale, df-invariante Un- terb¨undel zerf¨allt, genannt stabiles, instables und zentrales Unterb¨undel, so dass df Vektoren des stabilen Unterb¨undels st¨arker kontrahiert als Vektoren in zentrale Richtung sowie Vektoren in instabile Richtung st¨arker streckt als Vektoren in zentrale Richtung. Die stabilen und instabilen Unterb¨undel sind eindeutig integrabel zu f-invarianten stabilen und instabilen Bl¨atterungen, w¨ahrend das zentrale Unterb¨undel nicht einmal in einem schwachen Sinne integrabel sein muss. Diese Arbeit beschr¨ankt sich auf Systeme, die eine f-invariante Zentrumsbl¨atterung tangential zum zentralen Unterb¨undel be- sitzen. Die Eigenschaften der Zentrumsbl¨atterung sind eine M¨oglichkeit, um partiell hyperbolische Systeme zu klassifizieren, und dieser Weg soll in dieser Arbeit beschritten werden. Da sich die bekannten Beispielklassen vor allem hinsichtlich der Eigenschaften ihrer Zentrumsbl¨atterung unterscheiden, scheint dies eine sinnvolle Kategorisierung zu sein. Am Anfang dieser Arbeit stand die Idee, partiell hyperbolische Systeme auf hyperbolische Systeme zur¨uckzuf¨uhren, indem man sich der zentralen Richtung, die bei allen Beweisen Probleme bereitet, zu entledigen versucht. Mathematisch formuliert geht man zum Quotientenraum ¨uber, der entsteht, wenn man jeweils alle Punkte auf einer Zentrumsmannigfaltigkeit miteinan- der identifiziert. In diesem Quotientenraum kann man mit gutem Recht ein hyperbolisches Verhalten des induzierten Systems erwarten, auch wenn Glattheitseigenschaften auf dem Weg der Quotientenbildung verloren gehen. Im Allgemeinen, und hier liegt nun das Problem, ist der Quotientenraum einer Bl¨atterung in den meisten F¨allen nicht einmal ein Hausdorffraum. Doch die Hausdorffeigenschaft ist sicherlich ein Minimum dessen, was man von einem Raum verlangen muss, um in ihm sinnvoll die Dynamik eines Sys- tems untersuchen zu k¨onnen. Deshalb ist es notwendig zu fordern, dass alle Zentrumsmannigfaltigkeiten kompakt sind und endliche Holonomie besitzen. Was dies im Einzelnen ist und was es impliziert, wird im folgenden einleiten- den Kapitel ausf¨uhrlich dargelegt. An dieser Stelle soll es zun¨achst gen¨ugen festzuhalten, dass die Zentrumsbl¨atterung weitere zus¨atzliche Eigenschaften besitzen muss. Unter diesen Bedingungen induziert dann f eine Abbildung F auf dem Quotientenraum, die in einem topologischen Sinne viele hyperbo- lische Eigenschaften aufweist. Die Strategie, die den folgenden Beweisen zu- grunde liegt, ist, die hyperbolischen Eigenschaften immer dann auszunutzen, wenn die zentrale Richtung Schwierigkeiten bereitet, gleichzeitig aber auf f und die glatten invarianten Bl¨atterungen in M zur¨uckzugreifen, sobald der Beweis Glattheitseigenschaften erfordert. Die Annahme kompakter Zentrumsmannigfaltigkeiten scheint sehr restriktiv, 7 Vorwort ist aber, obwohl theoretisch motiviert, auch von praktischer Bedeutung, da eine wichtige Beispielklasse partiell hyperbolischer Systeme, die sogenannten Schiefprodukte, kompakte Zentrumsmannigfaltigkeiten besitzt. Die Schief- produkte sind in den letzten Jahren aus vielen Gr¨unden, wie z.B. Existenz von SRB-Maßen, intensiv studiert worden. Diese Arbeit ist nun ein Schritt hin zu einem besseren Verst¨andisses des Zusammenhanges zwischen partiell hyperbolischen Systemen mit kompakter Zentrumsbl¨atterung und Schiefpro- dukten bzw. endlichen Uberlagerungen¨ von Schiefprodukten. Im Falle einer dreidimensionalen Mannigfaltigkeit hat Pujals auf einer Konferenz im Jahr 2001 die Vermutung ge¨außert, dass beide identisch sind, die auch durch Ar- beiten von Bonatti und Wilkinson in [BW05a] in leicht abge¨anderter Form best¨arkt wird. Die folgenden drei Ergebnisse sind die wichtigsten, die in dieser Arbeit erzielt werden konnten: Ergebnisse: 1. Ein partiell hyperbolisches System f : M → M mit einer kompakten Zentrumsbl¨atterung mit endlicher Holonomie ist dynamisch koh¨arent, d.h. es existieren Bl¨atterungen zum zentralinstabilen und zum zentral- stabilen Unterb¨undel. Dieses Ergebnis findet sich als Theorem 1.24 im Kapitel 1. 2. Weiter zeigen wir das Beschattungslemma in Theorem 1.65 in Kapi- tel 1 f¨ur ein partiell hyperbolisches System f : M → M mit einer kompakten Zentrumsbl¨atterung mit endlicher Holonomie. 3. Es werde weiter angenommen, dass das instabile Unterb¨undel eindi- mensional ist. Dann kann ein partiell hyperbolisches System f : M → M mit einer kompakten Zentrumsbl¨atterung mit endlicher Holonomie auf einer zweifachen Uberlagerung¨ zu einem partiell hyperbolischen System f˜ hochgehoben werden, so dass die hochgehobene Zentrumsbl¨atterung nur triviale Holonomie
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