Global Analysis

http://dx.doi.org/10.1090/pspum/014

Proceedings of Symposia in

PURE MATHEMATICS

Volume 14

Global Analysis Shiing-Shen Chern Stephen Smale Editors

PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT THE UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA

JULY 1-26, 1968

Prepared by the American Mathematical Society under National Science Foundation Grant GP-8410

AMS 1968 Primary Subject Classification 3465

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109 8 7 6 54 96 95 94 93 92 91

CONTENTS Preface Bumpy Metrics

BY R. ABRAHAM

Nongenericity of Q-Stability . . . . BY R. ABRAHAM AND S. SMALE

On Certain Automorphisms of Nilpotent Lie Groups BY LOUIS AUSLANDER AND JOHN SCHEUNEMAN

Periodic Points of Anosov Diffeomorphisms . BY THOMAS F. BANCHOFF AND MICHAEL I. ROSEN

Topological Entropy and Axiom A BY ROBERT BOWEN

Zeta Functions of Restrictions of the Shift Transformation BY R. BOWEN AND O. E. LANFORD III

On the Generic Nature of Property HI for Hamiltonian Vector Fields BY MICHAEL A. BUCHNER

Locating Invariant Sets . . . . BY R. W. EASTON

Anosov Diffeomorphisms . . . . BY JOHN FRANKS

Endomorphisms of the Ripmann Sphere BY JOHN GUCKENHEIMER

Expanding Maps and Transformation Groups BY MORRIS W. HIRSCH

Stable Manifolds and Hyperbolic Sets BY MORRIS W. HIRSCH AND CHARLES C. PUGH

Commuting Diffeomorphisms BY NANCY KOPELL

A Generic Phenomenon in Conservative Hamiltonian Systems BY KENNETH R. MEYER AND JULIAN PALMORE

Nondensity of Axiom A(a) on S2

BY SHELDON E. NEWHOUSE

Nonsingular Endomorphisms of the Circle BY ZBIGNIEW NITECKI

A Note on Q-Stability BY J. PALIS

Structural Stability Theorems BY J. PALIS AND S. SMALE

A Global Approximation Theorem for Hamiltonian Systems BY R. CLARK ROBINSON

Strongly Mixing Transformations . . . . . BY RICHARD SACKSTEDER

v

CONTENTS

The Depth of the Center of 2-Manifolds BY A. J. SCHWARTZ AND E. S. THOMAS

Second Order Ordinary Differential Equations on Differentiable Manifolds BY S. SHAHSHAHANI

Expanding Maps . . . . . BY MICHAEL SHUB

Notes on Differentiable Dynamical Systems BY STEPHEN SMALE

The Q—Stability Theorem . . . . BY S. SMALE

Anosov Flows on Infra-Homogeneous Spaces. BY PER TOMTER

The "DA" Maps of Smale and Structural Stability BY R. F. WILLIAMS

The Zeta Function in Global Analysis BY R. F. WILLIAMS

Classification of One Dimensional Attractors . BY R. F. WILLIAMS

Author Index . . . . . . Subject Index . . . . . .

253

265

vi

PREFACE

The papers in these Proceedings grew out of lectures given at the fifteenth Summer Mathematical Institute of the American Mathematical Society, whose topic was global analysis. The Institute was held at the University of California at Berkeley from July 1 to July 26, 1968, and was partially financed by the National Science Foundation.

Notes of lectures were distributed at the time of the conference and some of the papers here are just as in those notes. These volumes, however, can be distinguished from the notes in the sense that in general the papers here are not just expositions of material that has or will appear elsewhere; most of the articles could just as well have appeared in Journals.

The unity given by the subject matter makes it desirable to collect them here. It is hoped that the volumes will provide an important start to the scientist who wishes to learn what is going on in that part of mathematics called global analysis.

The organizing committee for the institute consisted of: F. Browder, S.-S. Chern, L. Hormander, I. Singer, and S. Smale, with the co-editors serving as co-chairmen.

Seminar organizers were: F. Browder, E. Calabi, H. Goldschmidt, R. Hermann, C. Morrey, R. Palais, C. Pugh, I. Singer, and D. Spencer.

Finally the editors would like to thank the many people who made the institute and volumes possible. Of especially direct help to ourselves were Celeste Andrade, Ann Harrington, Gordon and Jacqueline Walker.

S.-S. Chern December 1968

S. Smale

vn

AUTHOR INDEX

Roman numbers refer to pages on which a reference is made to an author or a work of an author. Italic numbers refer to pages on which a complete reference to a work by the author is given. Boldface numbers indicate the first page of the articles in the book.

Abraham, Ralph, 1, 5, 5, 8, 23, 41, 51, 52, 54, 54, 92,163,189, 191, 202, 203, 205, 213, 220, 233,234,237,238,241,243,272,297,341,361

Adler, R. L., 23, 24, 38, 41 Anderson, R. D., 123 Andronov, 278 Anosov, D. V., 9,61,62, 92,163,245,252,283,

326 Arens, R., 131 Arnold, V. I., 163, 179,183 Artin, M., 163, 335, 336, 338 Auslander, Louis, 9, 75,63,76, 78, 92,125,131,

184, 245, 250, 251, 274, 275, 276, 301, 309, 312,313,322,323,324,526

Avez, A., 131, 163, 299

Bade, W., 156 Banchoff, Thomas F., 17 Bareto, A., 267, 272 Bialynicki-Birula, A., 312, 326 BirkhorT, G. D., 253, 264 Borel, A., 12, 92, 290, 317, 326 Bott, R., 3, 121,123 Bourbaki, N., 319, 326 Bowen, Robert, 23, 41, 43, 290, 336, 338, 338,

339 Brouwer, 205 Buchner, Michael A., 51

Chern,S.-S., 121,723 Chevalley, C, 319, 321, 326 Chow, W. L., 252, 252 Coddington, E., 183 Conley, Charles, 55 Conze, J. P., 27, 41

Dieudonne, 163 Deprit, A., 189

Easton, R. W„ 55 Epstein, D. B. A., 274, 276 Evens, L., 341

Fatou, P., 48, 49, 95, 97, 725 Fine, N., 174,183 Fomin, S. V., 299, 301, 308,327 Franks, John, 61, 125, 126, 757, 273, 275, 276,

276, 283, 299, 327 Frederickson, Paul, 254, 264 Furstenburg, H., 23, 41

Garding, L., 327 Gelfand, I. M., 299, 301, 308, 327 Gottschalk, W., 37, 725 Green, L., 301, 309, 312, 323, 324,326 Gromoll, D., 1,5 Guckenheimer, John, 41,95,203,204,215,216,

217,220,277

Haefliger, A., 81,82, 92 Hahn, F., 312, 326 Harish-Chandra, 326 Hartman, P., 92,163, 184, 231 Hedlund, G., 725 Hedlund, W., 184 Helgason,S., 315, 316,527 Helson, H., 216, 220 Henrard, J., 189 Hermann, R., 252, 252 Herstein, I. N., 567 Hille, E., 725 Hirsch, Morris, 8,27,41,63,65,67,76,92,125,

757, 133, 765, 167, 192, 199, 202, 222, 224, 227,273,274,290,291,297

Hochschild, G., 92, 310,527 Holmes, 765

Jacobson, M. B., 122 Jacobson, N., 327 Julia, C, 95, 97, 103, 725

Kelley, J., 54, 133 Keynes, H. B., 27, 37, 41 Klingenberg, W., 3 Konheim, A. G., 23, 24, 41 Kopell, Nancy, 165 Kripke, B., 167, 168 Kupka, I., 167, 170,233,239

Lam, P. F., 166 Lanford, O. E. HI, 36, 41,43, 336, 338, 339 Lang, S., 765, 272 Lattes, M., 103 Levine, H. L, 189 Levinson, N., 183

MacLane, S., 92 McAndrew, M. H., 23, 24, 38, 41

Mackey, G. W„ 248, 252, 325, 327 Maler, A., 254, 264

363

364 AUTHOR INDEX

Malcev, A., 15. 76, 92. 327 Malgrange, B., 54, 54 Margulis, G. A., 327 Marsden, J., J. 51, 52. 54, 54, 233, 234, 243. 272 Mather, John, 61. 125, 148, 155, 163 Mautner, F., 301,308, 327 Mazur, B., 335, 336, 338 Meyer, Kenneth R., 3, 3. 185, 235, 243, 335,

336, 339 Meyer, W., 1,3, 3 Milnor, J., 92, 93. 276, 276. 300, 317, 327 Montel, 97, 119 Montgomery, D., 127, 128, 131 Moore, C. C , 301. 322. 325, 327 Moore, R. U 167, 263, 264 Morse, Marstert^l, 278 Moser,J.,61, 135 Mostow, George D., 301, 312, 324, 327 Munkres, J. R., 93 Myers, S B . , 128, 131

Nachbin, L., 322, 327 Nagy, 163 Narasimhan, R., 53, 54 Nemytskii, V. V.. 264 Newhouse, Sheldon E., 191, 210, 220 Nitecki, Zbigniew, 203 Novikov, S. P., 85

O Meara, O. T., 320, 327

Palais, R., 2, 3, 167, 180 Palis, Jacob, 167, 182, 184, 221, 222. 223. 231.

280, 286. 287, 290. 338 Palmore, Julian, 185. 189. 243 Parry, W , 36, 41. 43 Peixoto, M., 166, 174. 184. 203. 204. 220. 235.

239, 243. 267, 272, 329. 334 Poincare. H., 121, 189. 189 Polya, G., 49 Pontrjagin. L., 93, 278 Pugh, Charles C , 8. 27, 41, 65, 92, 131, 133,

163. 191, 192. 199. 202. 204, 205, 220, 222, 224, 272. 290, 291, 297. 327. 334

Reeb. G., 93. 168, 184 Riesz, 163 Ritt. J. F.. 95, 97, 123 Robbin, J., 3. 8. 52, 54. 92, 163. 203. 205, 213,

220, 237. 238, 241, 243. 272 Robertson, J. B., 27. 37, 41 Robinson, R. Clark, 3. 3, 233. 243, 243 Rokhlin, V. A.. 23, 35, 41 Rosen, Michael I., 17 Rosenberg, H., 289, 297 Rosenlicht, M., 312, 326

Sacksteder, Richard, 93, 245 Saks, S., 247, 252

Sarason, D., 216, 220 Schenkman, E., 275, 276 Scheuneman, John, 9, J5, 251, 313, 326 Schlessinger, M., 49 Schwartz, A. J , 253 Schweigert, G. E., 174, 183 Selberg, A., 317,527 Seminaire S. Lie, 327 Shahshahani, S., 265 Sherman, 325 Shub, Michael, 27, 41, 64, 65, 66,69, 90, 91, 93,

95, 102,119,123, 125,127,131,135,167, 182, 203, 204, 212, 220, 273, 274, 276, 287, 290, 299, 336, 339, 341

Sinai, Ja. G., 34, 41, 245, 252 Smale, Stephen, 2, 5, 8, 15, 17, 21, 23, 24, 26,

27. 30, 34, 36, 37, 41, 49, 61, 63, 65, 67, 81, 89, 93, 95, 96, 123, 135, 155, 163, 167, 182, 184, 191. 192, 197, 202, 202, 220, 221, 222, 222, 223, 231, 233, 239, 272, 272, 273, 276, 111, 280, 287, 287, 289, 297, 299, 301, 327, 329, 330, 334, 335, 336, 337, 338, 339, 341, 361

Smith, P. A., 81 Solovay, R., 167 Spanier. E., 93 Steenrod, N., 128, 131 Stepanov. V. V., 264 Sternberg, S.. 121, 123, 163, 167, 168, 172, 173,

180, 182, 184 Szego, G., 49

Tamagawa, T., 327 Tate, 163 Thorn, R., 17.271,329 Thomas, E. S., 253 Tomter, Per, 299

Van Dantzig, D., 126, 131 Van der Waerden, B., 126, 131 Van Kampen, E. R., 184 von Neumann, J., 248, 252

Wallach, Nolan R., 301 Walters. P., 27 Walton, Robert, 54 Weil, 335 Weinstein, A., 3 Weyl, H., 123 Wilder, R., 123 Williams. R. F., 34,4/, 112. 113, 123, 135, 160,

161, 163, 191, 192, 202, 273, 276, 285, 287, 329.334,335, 339, Ml, 361

Wolf, Joseph A.. 92, 93, 276, 276, 300, 301, 305, 308,316,317,527

Zelinsky, Daniel, 329 Zippin, L., 127. 128, 131, 258, 264

SUBJECT INDEX

Absolutely continuous spectrum, 323 Admissible metric, 157 Adopted metric, 148 Affine algebraic group, 319 Afline transformations, 312, 313 Aggregate, 111 Algebra

Clifford, 300, 320 Dual, 14 Free nilpotent Lie, 10 Quaternion, 320

Algebraic groups, 310 Algebraic number field, 20 Almost conformal, 130 Almost invariant, 247 Anisotropic, 320, 321

Over Q, 321 Anosov, 283

Derived from (DA), 329 Anosov automorphism, 9 Anosov coverings, 62

Splitting, 67 Anosov diffeomorphism, 17, 61, 159, 302, 312,

313 N-induced, 302, 310, 313, 324, 326

Anosov flow, 299, 302, 305 G-induced, 301 (G, D-induced, 299, 301, 305, 313, 317, 319 Of "Mixed type", 321

Arithmetic groups, 319 Arithmetic subgroups, 301, 319, 320 Attractive, 135 Attractor, 341

1-dimensional, 160, 341 Axiom A, 5, 191,341

Baire property, 265 Basic sets, 289 Branched 1-manifold, 341, 343 Bumpy metric, 1 Bundles, 345

Contracting, 289 Expanding, 289

Canonical coordinates, 27 Cantor set

k-, 195 (^„/c2)-thick, 195,197 Two component, 194

Cartan-decomposition, 316 Cartan subalgebra, 314, 315, 321 Cayley hyperbolic plane, 316 Central sequence, 253

Characteristic polynomials, 351 Clifford algebra, 300, 320 Clifford group, 321 Clifford-Klein form, 314, 317 Closed orbit, 277

Elementary, 234 Generic, 234, 282 Periodic, 277

Conformal expanding maps, 130 Conjugate

Q, 5, 96, 285, 289 n,63 Semi-, 273 Topologically, 61, 96, 125, 273, 282, 342, 356,

360

Diffeomorphisms Anosov, 17, 61, 159, 302, 312, 313

JV-induced, 302, 310, 313, 324, 326 Axiom A, 5, 191, 341 Future stable, 287 Local stable, 133 Mixing, 249 No cycle property, 289 Nonwandering, 5, 96, 191, 253 n-conjugate, 5, 96, 285, 289 Q-explosion, 296 Q-stable, 5, 285, 289 Periodic point, 17 Poisson stable, 254 Structurally stable, 61, 96, 223, 230, 267, 278,

283, 302 Topologically conjugate, 61,96,125,273,283,

342, 356, 360 Toral, 17

Differentiable dynamical system, 277 Differential equation, second order ordinary,

265 Differentiation in a metric space, 247 Defining sequence, 194 Diophantine equation, 320 Dual algebra, 14

Equivalent G-, 303 (G, TK 304

Ergodic, 301, 322, 323, 324 Ergodic measure, 130 Expanding maps, 60, 90, 125, 273

Almost conformal, 130 Conformal, 130

Expansive homeomorphism, 26, 336 Exponential growth, 300, 313, 317, 319

365

366 SUBJECT INDEX

Fiber contraction theorem, 136 Fibonacci number, 17 Finite type, 36, 43, 338 Fixed point, 133, 135

Attractive, 135 Homoclinic points, 197 Lefschitz fixed point formula, 336

Flow, 253 Anosov, 299, 302, 305

G-induced, 301 (G, D-induced, 299, 301, 305, 313, 317, 319

G-induced, 302, 303 G-equivalent, 303

(G, FHnduced, 304, 305 (G, T)-equivalence, 304 Of "Mixed type", 321

Geodesic, 299, 308, 314 Recurrence, 253 Strongly mixing, 322 Suspended, 301 Wandering, 253 Weakly mixing, 301, 322, 323, 324

Frobenius Theorem, 251

Generalized solenoid, 329, 341, 343 Presentation, 341,342

Elementary, 342, 351 Generalized stable and unstable manifolds, 330 Geodesic flows, 299, 308, 314 Gradient, 277

H-elementary, 51 Graph transform, 134,141 Group

Affinc algebraic, 319 Algebraic, 310 Arithmetic, 319 Exponential growths, 300, 313, 317, 319 Isogeny, 319, 321 OftypeI,323

Hirsch-Plotkin radical, 275 Homoclinic points, 197 Hyperbolic, 96, 133, 279, 286, 302 Hyperbolic affine transformations, 300 Hyperbolic automorphisms, 300, 302, 312, 322 Hyperbolic set, 160, 161 Hyperbolic nilmanifold endomorphisms, 63 Hyperbolic space

Complex, 316 Quaternionic, 316 Real, 316

Hyperbolic structure, 5, 191, 289 Hyperbolic toral endomorphisms, 63

Infrahomogeneous spaces, 125 Infranil-expanding map, 274 Infranil-manifold, 125, 299, 302, 324 Intregral manifolds, 251 Invariant, 248

Inverse limit, 34 Isogeny, 319, 321

Isolated invariant set, 55 Isolating submanifold, 55 Iwasawa-decomposition, 316, 325

Kupka-Smale theorem, 202, 266

Ladder. 345 Ladder map, 341 Lebesgue spectrum, 301, 325, 326 Lefschitz fixed point formula, 336 Levi complement, 318 Lie algebras, free nilpotent, 10 Lie derivative, 251 Lipschitz inverse function theorem, 137 Lipschitzian, 248

Manifold Branched I-, 341, 343 Center of, 253 Generalized stable and unstable, 330 Infranil-, 125,299,302,324 Integral, 251 Isolating sub-, 55 Nil - , 17 Semilocal stable, of/", 98 Stable, 141 Unstable, 133, 141,279

Map Con formal expanding, 130 Equivalent, 342 Eventually constant, 359 Expanding, 64, 90, 125 Infranil-expanding, 274 Inverse limit, 34 Ladder, 341 Nil-expanding, 273, 274 Q-condition, 359 Shift, 341

Measure, 343, 354 Ergodic, 130 Spectral, 323, 325

Metric Admissible, 157 Adopted, 148 Bumpy, 1

Metrically splitting, 67 Mixing diffeomorphisms, 249

Nil-expanding map, 273, 274 Nil manifolds, 17 Nilradical, 322 No cycle property, 289 Nondegenerate critical point, 278 Nonwandering point, 5, 96, 191, 253 Nonwandering set, 285 Number field

Algebraic, 20 Totally real, 17

SUBJECT INDEX 367

Periodic point, 17 Poincare map, 269 Poincare upper half-plane, 314 Poisson stable, 254 Polynomial growth, 91 Presentation, 341, 342

Elementary, 342, 351 Principal characteristic multipliers, 233 Product neighborhood, 67 t>seudotransverse, 241

Quadratic space, 320 Quaternions, 320, 321 Quaternion algebra, 320

Recurrence, 253 Regular sequence tending to x, 247 Riemannian spaces, two-point homogeneous,

316

Semilocal stable manifold of/, 98 Semiconjugate, 273 Semidirect product, 302, 307. 310 Sequence

Central, 253 Defining, 194 /•gap, 194 a>( *)-limit point, 254 Regular, tending to x, 247

Set Basic, 289 Hyperbolic, 160 Isolated invariant, 55 fc-Cantor, 195 (*„ fc2)-thick Cantor, 197 Stable, 282 Subbasic, 5 Two component Cantor. 194

Shift class, 342, 356 Equivalence, 342

Shift equivalence, 349 Shift map, 281,341

Sub-shifts of finite type, 36, 43. 338 Unilateral, 130

Space Infrahomogeneous, 125 Complex hyperbolic, 316 Quadratic, 320 Quaternionic hyperbolic, 316 Real hyperbolic, 316 Symmetric, 314, 315, 316

Two-point homogeneous Riemannian, 316 Spectral measure, 323, 325 Spectrum

Absolutely continuous, 323 Lebesgue, 301, 325, 326

Stable Q-, 5, 285, 289 Structurally, 61, 96

Stable manifold, 141 Stable Manifold Theorem, 5

For a hyperbolic set, 149 For a point, 146

Stable sets, 282 Strongly mixed modulo, 245, 246, 247, 249

Structural stability, 329 Structurally stable, 223, 230, 267, 278,283, 302 Subbasic set, 5 Suspended flow, 301 Suspension, 302, 310 Symmetric space, 314, 315, 316

Tangency, one-sided, 201 Topologically conjugate, 61, 96, 125, 273, 283,

342, 356, 360 Topological entropy, 23 Topologically equivalent, 278 Toral difleomorphisms, 17 Totally real number field, 17 Transversality condition, 280

Strong, 287 Transversality theorem, 2

Density, 53 Tubular family, 224 Tubular families, system of, 224

Compatible, 224

Unilateral shift, 130 Unstable ideal, 309, 323 Unstable manifolds, 133,141, 279

Vector field Baire Property, 265 Property HI, 51,234 Property H2, 234 Topologically equivalent, 278

Weil zeta functions, 351 Whitney-Cr topology, 265

Z-subalgebra, 9 Zero-dimensional basic sets, 338 Zeta function, 34,43, 335