Differentiable dynamical systems

Smale, S.

doi:10.1090/S0002-9904-1967-11798-1

Differentiable dynamical systems

Author: S. Smale
Journal: Bull. Amer. Math. Soc. 73 (1967), 747-817
DOI: https://doi.org/10.1090/S0002-9904-1967-11798-1
MathSciNet review: 0228014
Full-text PDF

References | Additional Information

1. R. Abraham and J. Marsden, Foundations of mechanics, Benjamin, New York, 1967.
2. R. Abraham and J. Robbin, Transversal mappings and flows, Benjamin, New York, 1967. MR 240836
3. R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. MR 175106
4. R. L. Adler and M. H. McAndrew, The entropy of Chebyshev polynomials, Trans Amer. Math. Soc. 121 (1966), 236-241. MR 189005
5. R. L. Adler and R. Palais, Homeomorphic-conjugacy of automorphisms on the torus, Proc. Amer. Math. Soc. 16 (1965), 1222-1225. MR 193181
6. A. Andronov and L. Pontryagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR 14 (1937), 247-251.
7. D. V. Anosov, Roughness of geodesic flows on compact Riemannian manifolds of negative curvature, Soviet Math. Dokl. 3 (1962), 1068-1070. MR 143156
8. D. V. Anosov, Ergodic properties of geodesic flows on closed Riemannian manifolds of negative curvature, Soviet Math. Dokl. 4 (1963), 1153-1156. MR 163258
9. D. V. Anosov, Geodesic flows on compact Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967) = Proc. Steklov Math. Inst. (to appear). MR 224110
10. V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian, Russian Math. Surveys 18 (1963), 9-36. MR 163025
11. V. I. Arnold and A. Avez, Problèmes ergodiques de la mécanique classique, Gauthier-Villars, Paris, 1966. MR 209436
12. E. Artin and B. Mazur, On periodic points, Ann. of Math. (2) 81 (1965), 82-99. MR 176482
12a. L. Auslander, L. Green, F. Hahn et al., Flows on homogeneous spaces, Princeton Univ. Press, Princeton, N. J., 1963. MR 167569
13. A. Avez, Ergodic theory of dynamical systems, Vols. I, II, Univ. of Minneapolis, Minnesota, 1966, 1967.
14. P. Billingsley, Ergodic theory and information, Wiley, New York, 1965. MR 192027
15. G. D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Amer. Math Soc., Providence, R. I., 1927.
16. George D. Birkhoff, Surface transformations and their dynamical applications, Acta Math. 43 (1922), no. 1, 1–119. MR 1555175, https://doi.org/10.1007/BF02401754
17. George D. Birkhoff, On the periodic motions of dynamical systems, Acta Math. 50 (1927), no. 1, 359–379. MR 1555257, https://doi.org/10.1007/BF02421325
18. G. D. Birkhoff, Nouvelles recherches sur les systèmes dynamique, Mém. Pont. Acad. Sci. Novi Lyncaei 1 (1935), 85-216.
19. George D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199–300. MR 1501070, https://doi.org/10.1090/S0002-9947-1917-1501070-3
20. A. Borel, et al., Seminar on transformation groups, Princeton Univ. Press, Princeton, N. J., 1960. MR 116341
21. R. Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203-248. MR 89473
22. W. Boyce, Commuting functions with no common fixed point, Abstract 67T-218, Notices Amer. Math. Soc. 14 (1967), 280.
23. H. Cartan, Seminaire H. Cartan espaces fibrés et homotopie, Mimeographed, Paris, 1949-50.
24. E. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 69338
25. J. Dieudonné, Foundations of modern analysis, Academic Press, New York, 1960. MR 120319
26. A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology 4 (1965), 1-8. MR 193634
27. E. Dynkin, The maximal subgroups of the classical groups, Amer. Math. Soc. Transl. (2) 6 (1957), 245-379.
28. R. Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401-405. MR 101283
29. R. Ellis, The construction of minimal discrete flows, Amer. J. Math. 87 (1965), 564-574. MR 185589
30. L. E. Elsgolts, An estimate for the number of singular points of a dynamical system defined on a manifold, Amer. Math. Soc. Transl. (1) 5 (1962), 498-510 (originally 68 (1952)). MR 47263
31. F. B. Fuller, The existence of periodic points, Ann. of Math. (2) 57 (1953), 229-230. MR 52764
32. F. B. Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133-148. MR 209600
33. H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477-515. MR 157368
34. I. M. Gelfand and S. V. Fomin, Geodesic flows on manifolds of constant negative curvature, Amer. Math. Soc. Transl. (2) 1 (1955), 49-66. MR 73959
35. W. H. Gottschalk, Minimal sets: An introduction to topological dynamics, Bull. Amer. Math. Soc. 64 (1958), 336-351. MR 100048
35a. W. H. Gottschalk and G. A. Hedlund, Topological dynamics, Amer. Math. Soc. Colloq. Publ., vol. 36, Amer. Math. Soc., Providence, R. I., 1955. MR 74810
36. E. Grosswald, Topics from the theory of numbers, Macmillan, New York, 1966. MR 228408
37. V. Guillemin and S. Sternberg, On a conjecture of Palais and Smale, Trans. Amer. Math. Soc. (to appear).
38. B. Halpern, Fixed points for iterates, (to appear). MR 256383
39. P. Hartman, Ordinary differential equations, Wiley, New York, 1964. MR 171038
40. J. Hadamard, Les surfaces à courbures opposées e leur lignes géodèsiques, J. Math. Pures Appl. 4 (1898), 27-73.
41. A. Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa (3) 16 (1962), 367-397. MR 189060
42. G. Hedlund, The dynamics of geodesic flows, Bull. Amer. Math. Soc. 45 (1939), 241-246.
43. R. Hermann, Lie groups for physicists, Benjamin, New York, 1966. MR 213463
44. R. Hermann, The formal linearization of a semisimple Lie algebra of vector fields about a singular point, Trans. Amer. Math. Soc. (to appear). MR 217225
45. E. Hopf, Ergodic theory, Springer, Berlin, 1937.
46. E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261-304. MR 1464
47. W. Y. Hsiang, Remarks on differentiable actions of noncompact semi-simple Lie groups on Euclidean spaces, Amer. J. Math. (to appear). MR 193640
48. W. C. Hsiang, and W. Y. Hsiang, Differentiable actions of compact connected classical groups, Amer. J. Math. (to appear). MR 253367
49. J. P. Huneke, Two counterexamples to a conjecture on commuting continuous functions of the closed unit interval, Abstract 67T-231, Notices Amer. Math. Soc. 14 (1967), 284.
50. G. Julia, Mémoire sur l'iteration des fonctions rationnelles, J. Math. Pures Appl. 4 (1918), 47-245.
51. S. Kobayashi and J. Eells, Problems in differential geometry, Proc. Japan-United States Seminar in Differential Geometry, Kyoto, Japan 1965, Nippon Hyoransha, Tokyo, 1966, pp. 167-177.
52. A. N. Kolmogoroff, On conservation of conditionally periodic motions for a small charge in Hamilton's function, Dokl. Akad. Nauk SSSR 98 (1954), 527-530. (Russian) MR 68687
53. N. Kopell, Thesis, Univ. of California, Berkeley, 1967.
54. B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329-387. MR 142696
55. I. Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457-484. MR 165536
56. S. Lang, Introduction to differentiable manifolds, Wiley, New York, 1962. MR 155257
57. S. Lefschetz, Differential equations: Geometric theory, Interscience, New York, 1957. MR 153903
58. N. Levinson, A second order differential equation with singular solutions, Ann. of Math. (2) 50 (1949), 127-153. MR 30079
59. E. Lima, Commuting vector fields on S, Ann. of Math. (2) 81 (1965), 70-81. MR 171286
60. E. Lima, Common singularities of commuting vector fields on 2-manifolds, Comment. Math. Helv. 39 (1964), 97-110. MR 176459
61. A. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Transl. (1) 9 (1962), 276-307 (originally 39 (1951)). MR 39734
62. L. Markus, Structurally stable differential systems, Ann. of Math. (2) 73 (1961), 1-19. MR 132888
63. Y. Matsushima, On the discrete subgroups and homogeneous spaces of nilpotent Lie groups, Nagoya Math. J. 2 (1951), 95-110. MR 41144
64. F. Mautner, Geodesic flows on symmetric Riemann spaces, Ann. of Math. (2) 65 (1957), 416-431. MR 84823
65. K. Meyer, Energy functions for Morse-Smale systems, Amer. J. Math, (to appear). MR 239220
66. D. Montgomery and L. Zippen, Topological transformation groups, Interscience, New York, 1955. MR 84705
67. C. Moore, Distal affine transformation groups, Amer. J. Math (to appear). MR 232887
68. Marston Morse, The calculus of variations in the large, American Mathematical Society Colloquium Publications, vol. 18, American Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original. MR 1451874
69. Harold Marston Morse, A One-to-One Representation of Geodesics on a Surface of Negative Curvature, Amer. J. Math. 43 (1921), no. 1, 33–51. MR 1506428, https://doi.org/10.2307/2370306
70. J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. K1. II 1962, 1-20. MR 147741
71. J. Moser, Perturbation theory for almost periodic solutions for undamped nonlinear differential equations, Internat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 71-79. MR 147740
72. G. D. Mostow, The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math. (2) 52 (1950), 606-636. MR 48464
73. J. Nash, Real algebraic manifolds, Ann. of Math. (2) 56 (1952), 405-421. MR 50928
74. V. V. Nemytskii, Some modern problems in the qualitative theory of ordinary differential equations, Russian Math. Surveys 20 (1965), 1-35.
75. K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), 531-538. MR 64057
76. S. P. Novikov, The topology summer Institute, Seattle 1963, Russian Math. Surveys 20 (1965), 145-167.
77. S. P. Novikov, Smooth foliations on three-dimensional manifolds, Russian Math. Surveys 19 (1964) No. 6, 79-81. MR 173270
78. J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874-920. MR 5803
79. R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295-323. MR 126506
80. R. S. Palais, Equivalence of nearby differentiate actions of a compact group, Bull. Amer. Math. Soc. 67 (1961), 362-364. MR 130321
81. R. S. Palais and T. Stewart, Deformations of compact differentiate transformation groups, Amer. J. Math. 82 (1960), 935-937. MR 120652
82. J. Palis, Thesis, Univ. of California, Berkeley, 1967.
83. M. Peixoto, On structural stability, Ann. of Math. (2) 69 (1959), 199-222. MR 101951
84. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101-120. MR 142859
85. M. Peixoto, Symposium on Differential Equations and Dynamical Systems (Puerto Rico) Academic Press, New York, 1967. MR 216908
86. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential equations 3 (1967), 214-227. MR 209602
87. M. Peixoto and C. Pugh, Structurally stable systems on open manifolds are never dense, (to appear). MR 224936
88. Oskar Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), no. 1, 703–728 (German). MR 1545194, https://doi.org/10.1007/BF01194662
89. H. Poincaré, Sur les courbes dèfinies par des équations différentielles, J. Math. 4° serie 1885.
90. H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Vols. I, II, III, Paris, 1892, 1893, 1899; reprint, Dover, New York, 1957.
91. C. Pugh, The closing lemma, Amer. J. Math. (to appear) MR 226669
92. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. (to appear). MR 226670
93. C. Pugh, The closing lemma for Hamiltonian systems, Proc. Internat. Congress Math. at Moscow, 1966. MR 226669
94. G. Reeb, Sur certaines propriétés topologiques des trajectoires des systemes dynamiques, Acad. Roy. Belg. Cl. Sci. Mem. Coll. 8° 27 (1952), no. 9. MR 58202
95. L. Reizinš, On systems of differential equations satisfying Smales conditions, Latvijas PSR Zinatnu Akad. Vestis Fiz. Tehn. Zinatnu Ser. 1964, 57-61. (Russian) MR 212312
96. F. Riesz, and B. Nagy, Functional analysis, Ungar, New York, 1955.
97. H. Rosenberg, A generalization of Morse-Smale inequalities, Bull. Amer. Math. Soc. 70 (1964), 422-427. MR 160247
98. H. Rosenberg, Actions of R, Comment. Math. Helv. 41 (1966-67), 170-178. MR 206978
99. R. Sacksteder, Foliations and pseudo-groups, Amer. J. Math. 87 (1965), 79-102.
100. R. Sacksteder, Degeneracy of orbits of actions of Rn on a manifold, Comment. Math. Helv. 41 (1966-67), 1-9. MR 212835
101. O. F. G. Shilling, Arithmetical algebraic geometry, Harper and Row, New York, 1956.
102. A. J. Schwartz, A generalization of a Poincaré-Bendixon theorem to closed two-dimensional manifolds, Amer. J. Math. 85 (1963), 453-458. MR 155061
103. A. J. Schwartz, Common periodic points of commuting functions, Michigan Math. J.12 (1965), 353-355. MR 181996
104. A. Schwartz, Vector fields on a solid torus (to appear).
105. H. Seifert, Closed integral curves in 3-space and isotopic two-dimensional deformations, Proc. Amer. Math. Soc. 1 (1950), 287-302. MR 37508
106. A. Selberg, Harmonic-analysis and discontinuous groups in weakly symmetric Riemann spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. MR 88511
107. L. P. Shil'nikov, The existence of an enumerable set of periodic motions in the neighborhood of a homoclinic curve, Dokl. Akad. Nauk SSSR 172 (1967), 298-301 = Soviet Math. Dokl. 8 (1967), 102-106. MR 209603
108. M. Shub, Thesis, Univ. of California, Berkeley, 1967.
109. S. Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc.66 (1960), 43-49. MR 117745
110. S. Smale, On gradient dynamical systems, Ann. of Math. (2) 74 (1961), 199-206. MR 133139
111. S. Smale, On dynamical systems, Bol. Soc. Mat. Mexicana (2) 5 (1960), 195-198. MR 141855
112. S. Smale, Dynamical systems and the topological conjugacy problem for diffeomorphisms, Proc. Internat. Congress Math. (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 490-496. MR 176487
113. S. Smale, A structurally stable differentiate homeomorphism with an infinite number of periodic points, Proc. Internat. Sympos. Non-linear Vibrations, vol. II, 1961, Izdat. Akad. Nauk Ukrain SSR, Kiev, 1963. MR 160220
114. S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 97-116. MR 165537
115. S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N. J., 1965, pp. 63-80. MR 182020
116. S. Smale, Structurally stable systems are not dense, Amer. J. Math. 88 (1966), 491-496. MR 196725
117. S. Smale, Dynamical systems on n-dimensional manifolds, Symposium on differential equations and dynamical systems (Puerto Rico) Academic Press, New York, 1967.
118. J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds (to appear). MR 339279
119. P. R. Stein and S. M. Ulam, Non-linear transformations studies on electronic computers, Rozprawy Mat. 39 (1964). MR 169416
120. S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math.79 (1957), 787-789. MR 96853
121. S. Sternberg, On the structure of local homeomorphisms of Euclidean n-space. II, Amer. J. Math. 80 (1958), 623-631. MR 96854
122. S. Sternberg, The structure of local homeomorphisms, III, Amer. J. Math. 81 (1959), 578-604. MR 109853
123. S. Sternberg, Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, N. J., 1964. MR 193578
124. R. Thorn, Sur une partition en cellules associés à unefunction sur une variété, C. R. Acad. Sci. Paris 228 (1949), 973-975. MR 29160
125. A. Weil, Adeles and algebraic groups, Notes, Institute for Advanced Study, Princeton, N. J., 1961.
126. A. Weil, On discrete subgroups of Lie groups, II, Ann. of Math. (2) 75 (1962), 578-602. MR 137793
127. R. Williams, One dimensional non-wandering sets (to appear). MR 217808

Additional Information

DOI: https://doi.org/10.1090/S0002-9904-1967-11798-1