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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
International Standard Serial Number 0082-0717 International Standard Book Number 0-8218-1414-1
Library of Congress Catalog Number 70-95271
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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