Ergebnisse der Mathematik Volume 53 und ihrer Grenzgebiete. A Series of Modern Surveys in Mathematics. 3. Folge

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1 Ergebnisse der Mathematik Volume 53 und ihrer Grenzgebiete 3. Folge A Series of Modern Surveys in Mathematics Editorial Board G.-M. Greuel, Kaiserslautern M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollár, Princeton G. Laumon, Orsay H. W. Lenstra, Jr., Leiden S. Müller, Bonn J. Tits, Paris D. B. Zagier, Bonn G. Ziegler, Berlin Managing Editor R. Remmert, Münster

2 Vítor Araújo Maria José Pacifico Three-Dimensional Flows

3 Vítor Araújo Maria José Pacifico Instituto de Matemática Universidade Federal do Rio de Janeiro Rio de Janeiro Brazil ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISSN Library of Congress Control Number: Mathematics Subject Classification (2010): 34Cxx, 34Dxx, 37C10, 37C20, 37C40, 37C70, 37D05, 37D20, 37D25, 37D30, 37D45, 37D50 Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 Não sou nada. Nunca serei nada. Não posso querer ser nada. À parte isso, tenho em mim todos os sonhos do mundo. Álvaro de Campos. Tabacaria. Vítor dedica a Simone, Andreia, Manuela e Domingos. Maria José Pacifico dedica a Maria, Laura e Ricardo.

5 Foreword E.N. Lorenz s twelve pages note Deterministic nonperiodic flow, published in 1963 in the Journal of the Atmospheric Sciences [139], surely ranks among the most influential inputs from the experimental sciences in the history of 20th century Mathematics. While investigating the validity of linear regression models for weather forecast, Lorenz hit upon the observation that typical trajectories, even in very simple models, are unstable, or sensitive to small changes in the initial conditions. This observation had deep philosophical and practical implications. From a mathematician s perspective, Lorenz s paper set the challenge of describing and explaining on rigorous grounds the sensitivity phenomenon, and its stability under small modifications of the dynamical model. By that time, starting in the early sixties, the mathematical theory of hyperbolic dynamical systems was being developed by Steven Smale and his collaborators and students in the West, and by the Moscow school (Anosov, Arnold, Sinai and coworkers) in the Soviet Union. Hyperbolicity theory does provide a conceptual framework for understanding stable sensitive behavior, and has rapidly become a main paradigm in dynamical systems theory. However, the geometric models proposed by Afraimovich, Bykov, Shil nikov [2, 4] and Guckenheimer, Williams [98, 274] suggested that the Lorenz system is actually not hyperbolic. This state of perplexity lasted until the renewal of the theory of partial hyperbolicity in the mid nineties. A number of breakthroughs were obtained that led to a very complete theory of stable (or robust) behavior for flows in three-dimensional spaces, including both hyperbolic systems and Lorenz-type systems. It is this theory that the authors present in this book. Their text provides a much needed coherent presentation of one of the fastest developing subjects in recent mathematical research. Theirs was not an easy task: the material is very rich and widespread in a large number of papers; it is also quite recent, so that assessing the relevance of different results may be tricky. A particularly successful compromise between all these aspects was achieved, and I am sure this book will be a useful reference both for the expert working in the field and the student looking for an introduction to the subject. I also expect it vii

6 viii Foreword to play a significant role towards the extension of this theory to flows in arbitrary dimensions, which is currently under way. Enjoy your reading! Rio de Janeiro Marcelo Viana

7 Preface In this book we present the elements of a general theory for flows on threedimensional compact boundaryless manifolds, encompassing flows with equilibria accumulated by regular orbits. The main motivation for the development of this theory was the Lorenz system of equations whose numerical solution suggested the existence of a robust chaotic attractor with a singularity coexisting with regular orbits accumulating on it. More than three decades passed before the existence of the Lorenz attractor was rigorously established by Warwick Tucker with a computer-assisted proof in the year The difficulty in treating this kind of system is both conceptual and numerical. On the one hand, the presence of the singularity accumulated by regular orbits prevents this invariant set from being uniformly hyperbolic. On the other hand, solutions slow down as they pass near the saddle equilibria and so numerical integration errors accumulate without bound. Trying to address this problem, a successful approach was developed by Afraimovich-Bykov-Shil nikov and Guckenheimer-Williams independently, leading to the construction of a geometrical model displaying the main features of the behavior of the solutions of the Lorenz system of equations. In the 1990 s a breakthrough was obtained by Carlos Morales, Enrique Pujals and Maria José Pacifico following very original ideas developed by Ricardo Mañé during the proof of the C 1 -stability conjecture, providing a characterization of robustly transitive attractors for three-dimensional flows, of which the Lorenz attractor is an example. This characterization placed this class of attractors within the realm of a weak form of hyperbolicity: they are partially hyperbolic invariant sets with volume expanding central direction (or volume hyperbolic sets). Moreover robustly transitive attractors without singularities were proved to be uniformly hyperbolic. Thus these results extend the classical uniformly hyperbolic theory for flows with isolated singularities. Once this was established it is natural to try and understand the dynamical consequences of partial hyperbolicity with central volume expansion. It is well known that ix

8 x Preface uniform hyperbolicity has very precise implications for the dynamics, geometry and statistics of the invariant set. It is important to ascertain which properties are implied by this new weak form of hyperbolicity, known today as singular-hyperbolicity. Significant advances at the topological and ergodic level where recently obtained through the work of many authors which deserve a systematic presentation. This is the main motivation for writing these notes. We hope to provide a global perspective of this theory and make it easier for the reader to approach the growing literature on this subject. There have been several books and monographs on the subject of Dynamical Systems. But there are many distinct aspects which together make this book unique. First of all, this book treats mostly continuous time dynamical systems, instead of its discrete counterpart which is exhaustively treated in some of the other texts. Second, this book treats all the subjects from a mathematical perspective with proofs of most of the results included. Some of the proofs are done in a different way than those in the original papers because, once the theory is organized, it is possible to simplify many of the original proofs. We also extend many of the results about singular-hyperbolicity to higher dimensional flows, adding some new and natural hypotheses on the flow. The proofs about such extensions are also included. Third, this book is meant to be an advanced graduate textbook and not just a reference book or monograph on the subject. This aspect is reflected in the way the cover material is presented, with careful and complete proofs, and precise references to any topic in the book. Finally, there is not enough room (or time!) to cover all the topics in an advanced graduate course. This means that the book is not exhaustive: the main topics still constitute a very active area of research, but the book tries to treat the core concepts thoroughly and others enough so the reader will be prepared to read further on the subject and, we hope, also be prepared to contribute with new results on this theory. It is a pleasure to thank our co-authors Carlos Morales and Enrique Pujals who made definitive contributions and helped build the theory of singular-hyperbolicity. We also thank Ivan Aguilar for providing the figures of his MSc. thesis at UFRJ, and Serafin Bautista and Alfonso Artigue for having communicated to us some arguments which we include in this text. We are indebted to several PhD. students at IM-UFRJ who read previous versions of this text and pointed out to us several places where the presentation should be improved, among them Laura Senos, Regis Castijos and João Reis. We extend our acknowledgment to Marcelo Viana who, besides being our coauthor, encouraged us to write this text. We are thankful to DMP-FCUP, IM-UFRJ and IMPA for providing us with the necessary time and conditions to write this text. We also profited from partial financial support from CNPq, FAPERJ, PRONEX-Dynamical Systems (Brazil) and CMUP-FCT (Portugal). Finally we thank the scientific committee of the XXVI Brazilian Mathematical Colloquium for selecting our proposal for an earlier version of this text. Rio de Janeiro Vítor Araújo and Maria José Pacifico

9 Contents 1 Introduction OrganizationoftheText Preliminary Definitions and Results Fundamental Notions and Definitions Critical Elements, Non-wandering Points, Stable and UnstableSets LimitSets,Transitivity,AttractorsandRepellers Hyperbolic Critical Elements Topological Equivalence, Structural Stability Low Dimensional Flow Versus Chaotic Behavior One-Dimensional Flows Two-Dimensional Flows Three Dimensional Chaotic Attractors Hyperbolic Flows Hyperbolic Sets and Singularities Examples of Hyperbolic Sets and Axiom A Flows Expansiveness and Sensitive Dependence on Initial Conditions ChaoticSystems Expansive Systems BasicTools The Tubular Flow Theorem Transverse Sections and the Poincaré Return Map The Hartman-Grobman Theorem on Local Linearization The (Strong) Inclination Lemma (or λ-lemma) Homoclinic Classes, Transitiveness and Denseness of PeriodicOrbits TheClosingLemma The Connecting Lemma The Ergodic Closing Lemma APerturbationLemmaforFlows xi

10 xii Contents Generic Vector Fields and Lyapunov Stability The Linear Poincaré Flow Hyperbolic Splitting for the Linear Poincaré Flow Dominated Splitting for the Linear Poincaré Flow Incompressible Flows, Hyperbolicity and Dominated Splitting Ergodic Theory PhysicalorSRBMeasures GibbsMeasuresVersusSRBMeasures Stability Conjectures Singular Cycles and Robust Singular Attractors Singular Horseshoe A Singular Horseshoe Map A Singular Cycle with a Singular Horseshoe First Return Map The Singular Horseshoe Is a Partially Hyperbolic Set with Volume Expanding Central Direction Bifurcations of Saddle-Connections Saddle-Connection with Real Eigenvalues InclinationFlipandOrbitFlip Saddle-Focus Connection and Shil nikov Bifurcations Lorenz Attractor and Geometric Models Properties of the Lorenz System of Equations The Geometric Model The Geometric Lorenz Attractor Is a Partially Hyperbolic Set with Volume Expanding Central Direction Existence and Robustness of Invariant Stable Foliation RobustnessoftheGeometricLorenzAttractors The Geometric Lorenz Attractor Is a Homoclinic Class Robustness on the Whole Ambient Space No Equilibria Surrounded by Regular Orbits with Dominated Splitting Homogeneous Flows and Dominated Splitting Dominated Splitting over the Periodic Orbits Dominated Splitting over Regular Orbits from the Periodic Ones Bounded Angles on the Splitting over Hyperbolic Periodic Orbits Dominated Splitting for the Linear Poincaré Flow Along RegularOrbits Uniform Hyperbolicity for the Linear Poincaré Flow Subadditive Functions of the Orbits of a Flow and Exponential Growth

11 Contents xiii Uniform Hyperbolicity for the Linear Poincaré Flow on thewholemanifold Robust Transitivity and Singular-Hyperbolicity DefinitionsandStatementofResults Equilibria of Robust Attractors Are Lorenz-Like Robust Attractors Are Singular-Hyperbolic BriefSketchoftheProofs Higher Dimensional Analogues Singular-Attractor with Arbitrary Number of Expanding Directions The Notion of Sectionally Expanding Sets Homogeneous Flows and Sectionally Expanding Attractors Attractors and Isolated Sets for C 1 Flows ProofofSufficientConditionstoObtainAttractors Robust Singular Transitivity Implies Attractors or Repellers Attractors and Singular-Hyperbolicity Uniformly Dominated Splitting over the Periodic Orbits Dominated Splitting over a Robust Attractor Robust Attractors Are Singular-Hyperbolic Flow-Boxes Near Equilibria Uniformly Bounded Angle Between Stable and Center-UnstableDirectionsonPeriodicOrbits Singular-Hyperbolicity and Robustness Cross-Sections and Poincaré Maps StableFoliationsonCross-Sections Hyperbolicity of Poincaré Maps Adapted Cross-Sections Global Poincaré Return Map The One-Dimensional Piecewise Expanding Map Denseness of Periodic Orbits and the One-Dimensional Map Crossing Strips and the One-Dimensional Map HomoclinicClass SufficientConditionsforRobustness Denseness of Periodic Orbits and Transitivity with a Unique Singularity Unstable Manifolds of Periodic Orbits Inside Singular-Hyperbolic Attractors Expansiveness and Physical Measure StatementsoftheResultsandOverviewoftheArguments Robust Sensitiveness Existence and Uniqueness of a Physical Measure Expansiveness Proof of Expansiveness...208

12 xiv Contents Infinitely Many Coupled Returns Semi-global Poincaré Map A Tube-Like Domain Without Singularities EveryOrbitLeavestheTube The Poincaré Map Is Well-Defined on Σ j Expansiveness of the Poincaré Map Singular-Hyperbolicity and Chaotic Behavior Non-uniform Hyperbolicity TheStartingPoint The Hölder Property of the Projection Integrability of the Global Return Time Suspending Invariant Measures Physical Measure for the Global Poincaré Map Suspension Flow from the Poincaré Map Physical Measures for the Suspension PhysicalMeasurefortheFlow Hyperbolicity of the Physical Measure Absolutely Continuous Disintegration of the Physical Measure ConstructingtheDisintegration The Support Covers the Whole Attractor Singular-Hyperbolicity and Volume Dominated Decomposition and Zero Volume Dominated Splitting and Regularity Uniform Hyperbolicity Singular-Hyperbolicity and Zero Volume Partial Hyperbolicity and Zero Volume on C 1+ Flows PositiveVolumeVersusTransitiveAnosovFlows Zero-Volume for C 1 Generic Singular-Hyperbolic Attractors Extension to Sectionally Expanding Attractors in Higher Dimensions Global Dynamics of Generic 3-Flows Spectral Decomposition A Dichotomy for C 1 Generic 3-Flows Some Consequences of the Generic Dichotomy Generic 3-Flows, Lyapunov Stability and Singular- Hyperbolicity C 1 Generic Incompressible Flows Conservative Tubular Flow Theorem Realizable Linear Flows Blending Oseledets Directions Along an Orbit Segment Lowering the Norm: Local Procedure Lowering the Norm: Global Procedure Proof of the Dichotomy with Singularities (Theorem 9.4). 305

13 Contents xv 10 Related Results and Recent Developments More on Singular-Hyperbolicity Topological Dynamics AttractorsthatResembletheLorenzAttractor Unfolding of Singular Cycles Contracting Lorenz-Like Attractors Unfolding of Singular Cycles Dimension Theory, Ergodic and Statistical Properties LargeDeviationsfortheLorenzFlow Central Limit Theorem for the Lorenz Flow Decay of Correlations Decay of Correlations for the Return Map and Quantitative Recurrence on the Geometric Lorenz Flow Non-mixing Flows and Slow Decay of Correlations Decay of Correlations for Flows Thermodynamical Formalism Generic Conservative Flows in Dimension Appendix A Lyapunov Stability on Generic Vector Fields Appendix B A Perturbation Lemma for Flows Appendix C Robustness of Dominated Decomposition References Index...355

14 List of Figures 2.1 A sinkandasource Lorenz strange attractor A saddle singularity σ for bi-dimensional flow The flow near a hyperbolic saddle periodic orbit through p The equivalence relation defining the suspension flow of f over the roof function r Suspension flow over Anosov diffeomorphism with constant roof The solenoid attractor Sensitive dependence on initial conditions Linearization of orbits near a regular point of a flow Theinclinationlemma Closingarecurrentorbit The connecting lemma for C 1 flows Disintegration along centre-unstable leaves The holonomy maps A singular-horseshoe map A Smalehorseshoemap A singular cycle The vector field X Producing a unique tangency One point of tangency The final vector field The cross-section C s at p The first return map to Q The first return map at D u (p) The singular horseshoe return map The unstable curves of Ω F tangent at (α, 0) Breaking the saddle-connection a Inclination-flip. b Orbit-flip Saddle-focus connection Local stable and unstable manifolds near σ 0,σ 1 and σ 2, and the ellipsoid E xvii

15 xviii List of Figures 3.17 The trapping bi-torus The evolution of a generic orbit inside U Another view of the Lorenz attractor Behavior near the origin R takes Σ ± to S The return map image P(S ) Projection on I through the stable leaves and a sketch of the image of one leaf under the return map The Lorenz map f The field η φ Y The field F and the parallel condition The orbit of q under the linear flow, the vector field at different points of this orbit and the relative position with the image of the vector w given by the angle θ A non-robustly transitive Lorenz transformation A sketch of the construction of a robust singular-attractor in higher dimensions Cross-sections near a Lorenz-like equilibrium The sections Σ, Σ(γ), the manifolds W s (x), W ss (x), W s (x, Σ) and the projection R Σ, on the right. On the left, the square [0, 1] 2 is identified with Σ through the map h, where FΣ s becomes the horizontal foliation and γ is a transverse curve, and solid lines with arrows indicate the flow direction The stable manifolds on the cross-section and the cu-curve γ connecting them An adapted cross-section for Λ The construction of a δ-adapted cross-section for a regular x Λ The stable manifold of σ, the unstable manifolds of p,q and the points in Σ Definition of W u,+ and W u, The unperturbed flow X The perturbed flow Y The arc J and cross-sections Σ p,σ i, How I accumulates D u (x ) Distances near a point in the stable-manifold Relative positions of the strong-stable manifolds and orbits A tube-like domain Entering the flow-box of a singularity Exiting the tube at Σ j+1 flowing from Σ j Expansion within the tube Center-unstable leaves on the suspension flow Hyperbolic times and projections Leaves crossing F s (x 0,δ 2 ) and the projection p Transitiveness and support of the physical measure A modified geometric Lorenz attractor

16 List of Figures xix 9.2 A sketch of the construction of a singular-hyperbolic attractor which is not the disjoint union of homoclinic classes The accumulation on one of the components of W s (σ ) The conservative change of coordinates straightening out all orbits Realizing vector fields given a linear map L t The one-dimensional map for the contracting Lorenz model A double homoclinic connection A correlation function for a non-mixing flow