Differentiable dynamical systems
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-
1. R. Abraham and J. Marsden, Foundations of mechanics, Benjamin, New York, 1967.
- Ralph Abraham and Joel Robbin, Transversal mappings and flows, W. A. Benjamin, Inc., New York-Amsterdam, 1967. An appendix by Al Kelley. MR 0240836
- R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9
- R. L. Adler and M. H. McAndrew, The entropy of Chebyshev polynomials, Trans. Amer. Math. Soc. 121 (1966), 236–241. MR 189005, DOI 10.1090/S0002-9947-1966-0189005-0
- R. L. Adler and R. Palais, Homeomorphic conjugacy of automorphisms on the torus, Proc. Amer. Math. Soc. 16 (1965), 1222–1225. MR 193181, DOI 10.1090/S0002-9939-1965-0193181-8 6. A. Andronov and L. Pontryagin, Systèmes grossiers, Dokl. Akad. Nauk. SSSR 14 (1937), 247-251.
- D. V. Anosov, Roughness of geodesic flows on compact Riemannian manifolds of negative curvature, Dokl. Akad. Nauk SSSR 145 (1962), 707–709 (Russian). MR 0143156
- D. V. Anosov, Ergodic properties of geodesic flows on closed Riemannian manifolds of negative curvature, Dokl. Akad. Nauk SSSR 151 (1963), 1250–1252 (Russian). MR 0163258
- D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
- V. I. Arnol′d, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 13–40 (Russian). MR 0163025
- V. I. Arnold and A. Avez, Problèmes ergodiques de la mécanique classique, Monographies Internationales de Mathématiques Modernes, vol. 9, Gauthier-Villars, Éditeur, Paris, 1967 (French). MR 0209436
- M. Artin and B. Mazur, On periodic points, Ann. of Math. (2) 81 (1965), 82–99. MR 176482, DOI 10.2307/1970384
- L. Auslander, L. Green, and F. Hahn, Flows on homogeneous spaces, Annals of Mathematics Studies, No. 53, Princeton University Press, Princeton, N.J., 1963. With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. MR 0167569, DOI 10.1515/9781400882021 13. A. Avez, Ergodic theory of dynamical systems, Vols. I, II, Univ. of Minneapolis, Minnesota, 1966, 1967.
- Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027 15. G. D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Amer. Math Soc., Providence, R. I., 1927.
- George D. Birkhoff, Surface transformations and their dynamical applications, Acta Math. 43 (1922), no. 1, 1–119. MR 1555175, DOI 10.1007/BF02401754
- George D. Birkhoff, On the periodic motions of dynamical systems, Acta Math. 50 (1927), no. 1, 359–379. MR 1555257, DOI 10.1007/BF02421325 18. G. D. Birkhoff, Nouvelles recherches sur les systèmes dynamique, Mém. Pont. Acad. Sci. Novi Lyncaei 1 (1935), 85-216.
- George D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199–300. MR 1501070, DOI 10.1090/S0002-9947-1917-1501070-3
- Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
- Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203–248. MR 89473, DOI 10.2307/1969996 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.
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
- J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
- Albrecht Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology 4 (1965), 1–8. MR 193634, DOI 10.1016/0040-9383(65)90044-3 27. E. Dynkin, The maximal subgroups of the classical groups, Amer. Math. Soc. Transl. (2) 6 (1957), 245-379.
- Robert Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–405. MR 101283, DOI 10.2140/pjm.1958.8.401
- Robert Ellis, The construction of minimal discrete flows, Amer. J. Math. 87 (1965), 564–574. MR 185589, DOI 10.2307/2373063
- L. E. Èl′sgol′c, An estimate for the number of singular points of a dynamical system defined on a manifold, Amer. Math. Soc. Translation 1952 (1952), no. 68, 14. MR 0047263
- F. B. Fuller, The existence of periodic points, Ann. of Math. (2) 57 (1953), 229–230. MR 52764, DOI 10.2307/1969856
- F. Brock Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133–148. MR 209600, DOI 10.2307/2373103
- H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477–515. MR 157368, DOI 10.2307/2373137
- I. M. Gel′fand and S. V. Fomin, Geodesic flows on manifolds of constant negative curvature, Amer. Math. Soc. Transl. (2) 1 (1955), 49–65. MR 0073959, DOI 10.1090/trans2/001/04
- W. H. Gottschalk, Minimal sets: an introduction to topological dynamics, Bull. Amer. Math. Soc. 64 (1958), 336–351. MR 100048, DOI 10.1090/S0002-9904-1958-10223-2
- Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R.I., 1955. MR 0074810, DOI 10.1090/coll/036
- Emil Grosswald, Topics from the theory of numbers, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0228408 37. V. Guillemin and S. Sternberg, On a conjecture of Palais and Smale, Trans. Amer. Math. Soc. (to appear).
- Benjamin Halpern, Fixed points for iterates, Pacific J. Math. 25 (1968), 255–275. MR 256383, DOI 10.2140/pjm.1968.25.255
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038 40. J. Hadamard, Les surfaces à courbures opposées e leur lignes géodèsiques, J. Math. Pures Appl. 4 (1898), 27-73.
- André Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 16 (1962), 367–397 (French). MR 189060 42. G. Hedlund, The dynamics of geodesic flows, Bull. Amer. Math. Soc. 45 (1939), 241-246.
- Robert Hermann, Lie groups for physicists, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0213463
- Robert Hermann, The formal linearization of a semisimple Lie algebra of vector fields about a singular point, Trans. Amer. Math. Soc. 130 (1968), 105–109. MR 217225, DOI 10.1090/S0002-9947-1968-0217225-7 45. E. Hopf, Ergodic theory, Springer, Berlin, 1937.
- Eberhard Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 91 (1939), 261–304 (German). MR 1464
- Wu-yi Hsiang, On the classification of differentiable $\textrm {SO}(n)$ actions on simply connected $\pi$-manifolds, Amer. J. Math. 88 (1966), 137–153. MR 193640, DOI 10.2307/2373051
- W. Hsiang, Remarks on differentiable actions of non-compact semi-simple Lie groups on euclidean spaces, Amer. J. Math. 90 (1968), 781–788. MR 253367, DOI 10.2307/2373483 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.
- A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), 527–530 (Russian). MR 0068687 53. N. Kopell, Thesis, Univ. of California, Berkeley, 1967.
- Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
- Ivan Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457–484 (French). MR 165536
- Serge Lang, Introduction to differentiable manifolds, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155257
- Solomon Lefschetz, Differential equations: Geometric theory, 2nd ed., Pure and Applied Mathematics, Vol. VI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0153903
- Norman Levinson, A second order differential equation with singular solutions, Ann. of Math. (2) 50 (1949), 127–153. MR 30079, DOI 10.2307/1969357
- Elon L. Lima, Commuting vector fields on $S^{3}$, Ann. of Math. (2) 81 (1965), 70–81. MR 171286, DOI 10.2307/1970383
- Elon L. Lima, Common singularities of commuting vector fields on $2$-manifolds, Comment. Math. Helv. 39 (1964), 97–110. MR 176459, DOI 10.1007/BF02566947
- A. I. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Translation 1951 (1951), no. 39, 33. MR 0039734
- Lawrence Markus, Structurally stable differential systems, Ann. of Math. (2) 73 (1961), 1–19. MR 132888, DOI 10.2307/1970280
- Yozô Matsushima, On the discrete subgroups and homogeneous spaces of nilpotent Lie groups, Nagoya Math. J. 2 (1951), 95–110. MR 41144, DOI 10.1017/S0027763000010096
- F. I. Mautner, Geodesic flows on symmetric Riemann spaces, Ann. of Math. (2) 65 (1957), 416–431. MR 84823, DOI 10.2307/1970054
- K. R. Meyer, Energy functions for Morse Smale systems, Amer. J. Math. 90 (1968), 1031–1040. MR 239220, DOI 10.2307/2373287
- Deane Montgomery, Topological transformation groups, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 185–188. MR 0084705
- Calvin C. Moore, Distal affine transformation groups, Amer. J. Math. 90 (1968), 733–751. MR 232887, DOI 10.2307/2373481
- 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, DOI 10.1090/coll/018
- 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, DOI 10.2307/2370306
- J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1–20. MR 147741
- Jürgen 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 0147740
- George Daniel Mostow, The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math. (2) 52 (1950), 606–636. MR 48464, DOI 10.2307/1969437
- John Nash, Real algebraic manifolds, Ann. of Math. (2) 56 (1952), 405–421. MR 50928, DOI 10.2307/1969649 74. V. V. Nemytskii, Some modern problems in the qualitative theory of ordinary differential equations, Russian Math. Surveys 20 (1965), 1-35.
- Katsumi Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), 531–538. MR 64057, DOI 10.2307/1969716 76. S. P. Novikov, The topology summer Institute, Seattle 1963, Russian Math. Surveys 20 (1965), 145-167.
- S. P. Novikov, Smooth foliations on three-dimensional manifolds, Uspehi Mat. Nauk 19 (1964), no. 6 (120), 89–91 (Russian). MR 0173270
- J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874–920. MR 5803, DOI 10.2307/1968772
- Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 126506, DOI 10.2307/1970335
- Richard S. Palais, Equivalence of nearby differentiable actions of a compact group, Bull. Amer. Math. Soc. 67 (1961), 362–364. MR 130321, DOI 10.1090/S0002-9904-1961-10617-4
- Richard S. Palais and Thomas E. Stewart, Deformations of compact differentiable transformation groups, Amer. J. Math. 82 (1960), 935–937. MR 120652, DOI 10.2307/2372950 82. J. Palis, Thesis, Univ. of California, Berkeley, 1967.
- M. M. Peixoto, On structural stability, Ann. of Math. (2) 69 (1959), 199–222. MR 101951, DOI 10.2307/1970100
- M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120. MR 142859, DOI 10.1016/0040-9383(65)90018-2
- Jack K. Hale and Joseph P. LaSalle (eds.), Differential equations and dynamical systems, Academic Press, New York-London, 1967. MR 0216908
- M. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214–227. MR 209602, DOI 10.1016/0022-0396(67)90026-5
- M. M. Peixoto and C. C. Pugh, Structurally stable systems on open manifolds are never dense, Ann. of Math. (2) 87 (1968), 423–430. MR 224936, DOI 10.2307/1970713
- Oskar Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), no. 1, 703–728 (German). MR 1545194, DOI 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.
- Charles C. Pugh, The closing lemma, Amer. J. Math. 89 (1967), 956–1009. MR 226669, DOI 10.2307/2373413
- Charles C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010–1021. MR 226670, DOI 10.2307/2373414
- Charles C. Pugh, The closing lemma, Amer. J. Math. 89 (1967), 956–1009. MR 226669, DOI 10.2307/2373413
- Georges Reeb, Sur certaines propriétés topologiques des trajectoires des systèmes dynamiques, Acad. Roy. Belg. Cl. Sci. Mém. Coll. in 8$^\circ$ 27 (1952), no. 9, 64 (French). MR 58202
- L. È. Reĭzin′, Systems of differential equations satisfying Smale’s conditions, Latvijas PSR Zinātņ. Akad. Vēstis Fiz. Tehn. Zinātņ. Sēr. 1964 (1964), no. 4, 57–61 (Russian, with English and Latvian summaries). MR 212312 96. F. Riesz, and B. Nagy, Functional analysis, Ungar, New York, 1955.
- Harold Rosenberg, A generalization of Morse-Smale inequalities, Bull. Amer. Math. Soc. 70 (1964), 422–427. MR 160247, DOI 10.1090/S0002-9904-1964-11126-5
- Harold Rosenberg, Actions of $R^{n}$ on manifolds, Comment. Math. Helv. 41 (1966/67), 170–178. MR 206978, DOI 10.1007/BF02566874 99. R. Sacksteder, Foliations and pseudo-groups, Amer. J. Math. 87 (1965), 79-102.
- Richard Sacksteder, Degeneracy of orbits of actons of $R^{m}$ on a manifold, Comment. Math. Helv. 41 (1966/67), 1–9. MR 212835, DOI 10.1007/BF02566864 101. O. F. G. Shilling, Arithmetical algebraic geometry, Harper and Row, New York, 1956.
- Arthur J. Schwartz, A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds, Amer. J. Math. 85 (1963), 453-458; errata, ibid 85 (1963), 753. MR 0155061, DOI 10.2307/2373120
- A. J. Schwartz, Common periodic points of commuting functions, Michigan Math. J. 12 (1965), 353–355. MR 181996, DOI 10.1307/mmj/1028999371 104. A. Schwartz, Vector fields on a solid torus (to appear).
- Herbert Seifert, Closed integral curves in $3$-space and isotopic two-dimensional deformations, Proc. Amer. Math. Soc. 1 (1950), 287–302. MR 37508, DOI 10.1090/S0002-9939-1950-0037508-5
- A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
- L. P. Šil′nikov, Existence of a countable set of periodic motions in a neighborhood of a homoclinic curve, Dokl. Akad. Nauk SSSR 172 (1967), 298–301 (Russian). MR 0209603 108. M. Shub, Thesis, Univ. of California, Berkeley, 1967.
- Stephen Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), 43–49. MR 117745, DOI 10.1090/S0002-9904-1960-10386-2
- Stephen Smale, On gradient dynamical systems, Ann. of Math. (2) 74 (1961), 199–206. MR 133139, DOI 10.2307/1970311
- Stephen Smale, On dynamical systems, Bol. Soc. Mat. Mexicana (2) 5 (1960), 195–198. MR 141855
- S. Smale, Dynamical systems and the topological conjugacy problem for diffeomorphisms, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 49–496. MR 0176487
- S. Smale, A structurally stable differentiable homeomorphism with an infinite number of periodic points, Qualitative methods in the theory of non-linear vibrations (Proc. Internat. Sympos. Non-linear Vibrations, Vol. II, 1961) Izdat. Akad. Nauk Ukrain. SSR, Kiev, 1963, pp. 365–366. MR 0160220
- Ivan Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457–484 (French). MR 165536
- Stephen Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 63–80. MR 0182020
- S. Smale, Structurally stable systems are not dense, Amer. J. Math. 88 (1966), 491–496. MR 196725, DOI 10.2307/2373203 117. S. Smale, Dynamical systems on n-dimensional manifolds, Symposium on differential equations and dynamical systems (Puerto Rico) Academic Press, New York, 1967.
- J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 5–46. MR 339279, DOI 10.1007/BF02684365
- P. R. Stein and S. M. Ulam, Non-linear transformation studies on electronic computers, Rozprawy Mat. 39 (1964), 66. MR 169416
- Shlomo Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809–824. MR 96853, DOI 10.2307/2372437
- Shlomo Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. II, Amer. J. Math. 80 (1958), 623–631. MR 96854, DOI 10.2307/2372774
- Shlomo Sternberg, The structure of local homeomorphisms. III, Amer. J. Math. 81 (1959), 578–604. MR 109853, DOI 10.2307/2372915
- Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578
- René Thom, Sur une partition en cellules associée à une fonction sur une variété, C. R. Acad. Sci. Paris 228 (1949), 973–975 (French). MR 29160 125. A. Weil, Adeles and algebraic groups, Notes, Institute for Advanced Study, Princeton, N. J., 1961.
- André Weil, On discrete subgroups of Lie groups. II, Ann. of Math. (2) 75 (1962), 578–602. MR 137793, DOI 10.2307/1970212
- R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473–487. MR 217808, DOI 10.1016/0040-9383(67)90005-5
Additional Information
- Journal: Bull. Amer. Math. Soc. 73 (1967), 747-817
- DOI: https://doi.org/10.1090/S0002-9904-1967-11798-1
- MathSciNet review: 0228014