Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Instability in $ {\rm Diff}\sp{r}$ $ (T\sp{3})$ and the nongenericity of rational zeta functions


Author: Carl P. Simon
Journal: Trans. Amer. Math. Soc. 174 (1972), 217-242
MSC: Primary 58F20
DOI: https://doi.org/10.1090/S0002-9947-1972-0317356-8
MathSciNet review: 0317356
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the search for an easily-classified Baire set of diffeomorphisms, all the studied classes have had the property that all maps close enough to any diffeomorphism in the class have the same number of periodic points of each period. The author constructs an open subset U of $ {\text{Diff}^r}({T^3})$ with the property that if f is in U there is a g arbitrarily close to f and an integer n such that $ {f^n}$ and $ {g^n}$ have a different number of fixed points. Then, using the open set U, he illustrates that having a rational zeta function is not a generic property for diffeomorphisms and that $ \Omega $-conjugacy is an ineffective means for classifying any Baire set of diffeomorphisms.


References [Enhancements On Off] (What's this?)

  • [1] R. Abraham, J. Robbin and A. Kelley, Transversal mappings and flows, Benjamin, New York, 1967. MR 39 #2181. MR 0240836 (39:2181)
  • [2] R. Abraham and S. Smale, Nongenericity of $ \Omega $-stability, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 5-8. MR 42 #6867. MR 0271986 (42:6867)
  • [3] M. Artin and B. Mazur, On periodic points, Ann of Math. (2) 81 (1965), 82-99. MR 31 #754. MR 0176482 (31:754)
  • [4] R. Bowen, Topological entropy and Axiom A, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 23-41. MR 41 #7066. MR 0262459 (41:7066)
  • [5] R. Bowen and O. E. Lanford III, Zeta functions of restrictions of the shift transformation, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 43-49. MR 42 #6284. MR 0271401 (42:6284)
  • [6] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/72), pp. 193-226. MR 4, 4313. MR 0287106 (44:4313)
  • [7] J. Guckenheimer, $ Axiom$   A$ + no\;cycles \Rightarrow {\zeta _f}(t)$ rational, Bull. Amer. Math. Soc. 76 (1970), 592-594. MR 40 #8078. MR 0254871 (40:8078)
  • [8] A. Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa (3) 16 (1962), 367-397. MR 32 #6487. MR 0189060 (32:6487)
  • [9] M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 133-163. MR 42 #6872. MR 0271991 (42:6872)
  • [10] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc. 76 (1970), 1015-1019. MR 0292101 (45:1188)
  • [11] M. Hirsch, J. Palis, C. Pugh and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/70), 121-134. MR 41 #7232. MR 0262627 (41:7232)
  • [12] K. R. Meyer, On the convergence of the zeta function for flows and diffeomorphisms, J. Differential Equations 5 (1969), 338-345. MR 38 #1701. MR 0233379 (38:1701)
  • [13] S. Newhouse, Nondensity of axiom $ {\text{A}}({\text{a}})$ on $ {S^2}$, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 191-202. MR 43 #2742. MR 0277005 (43:2742)
  • [14] Z. Nitecki, Differentiable dynamics, M.I.T. Press, Cambridge, Mass., 1972.
  • [15] J. Palis and S. Smale, Structural stability theorems, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 223-231. MR 42 #2505. MR 0267603 (42:2505)
  • [16] M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214-227. MR 35 #499. MR 0209602 (35:499)
  • [17] J. Robbin, On structural stability, Bull. Amer. Math. Soc. 76 (1970), 723-726. MR 41 #6235. MR 0261622 (41:6235)
  • [18] R. Sacker, On periodic solutions near homoclinic points, Mimeographed notes, Courant Institute, New York Univ., New York, 1965.
  • [19] M. Shub, Periodic orbits of hyperbolic diffeomorphisms and flows, Bull. Amer. Math. Soc. 75 (1969), 57-58. MR 38 #2815. MR 0234498 (38:2815)
  • [20] S. Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), 43-49. MR 22 #8519. MR 0117745 (22:8519)
  • [21] -, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 97-116. MR 29 #2818b. MR 0165537 (29:2818b)
  • [22] -, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Sympos. in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965, pp. 63- 80. MR 31 #6244. MR 0182020 (31:6244)
  • [23] -, Structurally stable systems are not dense, Amer. J. Math. 88 (1960), 491-496. MR 33 #4911. MR 0196725 (33:4911)
  • [24] -, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 37 #3598. MR 0228014 (37:3598)
  • [25] -, The $ \Omega $-stability theorem, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 289-297. MR 42 #6852. MR 0271971 (42:6852)
  • [26] -, Stability and genericity in dynamical systems, Séminaire Bourbaki Exposé 374, Lecture Notes in Math., vol. 180, Springer-Verlag, Berlin and New York, 1971, pp. 1-9. MR 42 #7461.
  • [27] R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473-487. MR 36 #897. MR 0217808 (36:897)
  • [28] -, The zeta function of an attractor, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 155-161. MR 38 #3877. MR 0235573 (38:3877)
  • [29] -, Zeta function in global analysis, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 335-339. MR 42 #1159. MR 0266252 (42:1159)
  • [30] -, The ``DA'' maps of Smale and structural stability, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 329-334. MR 41 #9296. MR 0264705 (41:9296)
  • [31] M. Hirsch, Foliations and non-compact transformation groups, Bull. Amer. Math. Soc. 76 (1970), 1020-1023. MR 0292102 (45:1189)
  • [32] C. Simon, On a classification of a Baire set of diffeomorphisms, Bull. Amer. Math. Soc. 77 (1971), 783-787. MR 0285018 (44:2242)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F20

Retrieve articles in all journals with MSC: 58F20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0317356-8
Keywords: Differentiable dynamical systems, zeta functions for diffeomorphisms, periodic point, stable manifold, foliation, generic property
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society