Instability in 𝐷𝑖𝑓𝑓^{𝑟} (𝑇³) and the nongenericity of rational zeta functions

Simon, Carl P.

doi:10.1090/S0002-9947-1972-0317356-8

Instability in $\textrm {Diff}^{r}$ $(T^{3})$ and the nongenericity of rational zeta functions
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Trans. Amer. Math. Soc. 174 (1972), 217-242 Request permission

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

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. Abraham and S. Smale, Nongenericity of $\Omega$-stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 5–8. MR 0271986
M. Artin and B. Mazur, On periodic points, Ann. of Math. (2) 81 (1965), 82–99. MR 176482, DOI 10.2307/1970384
Rufus Bowen, Topological entropy and axiom $\textrm {A}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 23–41. MR 0262459
R. Bowen and O. E. Lanford III, Zeta functions of restrictions of the shift transformation, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 43–49. MR 0271401
Neil Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/72), 193–226. MR 287106, DOI 10.1512/iumj.1971.21.21017
John Guckenheimer, Axiom $\textrm {A}+\textrm {no cycles}\Rightarrow \zeta _{f}\,(t)$ rational, Bull. Amer. Math. Soc. 76 (1970), 592–594. MR 254871, DOI 10.1090/S0002-9904-1970-12449-1
André Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 16 (1962), 367–397 (French). MR 189060
Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR 0271991
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc. 76 (1970), 1015–1019. MR 292101, DOI 10.1090/S0002-9904-1970-12537-X
M. Hirsch, J. Palis, C. Pugh, and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/70), 121–134. MR 262627, DOI 10.1007/BF01404552
K. R. Mayer, On the convergence of the zeta function for flows and diffeomorphisms, J. Differential Equations 5 (1969), 338–345. MR 233379, DOI 10.1016/0022-0396(69)90048-5
Sheldon E. Newhouse, Nondensity of axiom $\textrm {A}(\textrm {a})$ on $S^{2}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 191–202. MR 0277005

Differentiable dynamics

J. Palis and S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 223–231. MR 0267603
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
J. W. Robbin, On structural stability, Bull. Amer. Math. Soc. 76 (1970), 723–726. MR 261622, DOI 10.1090/S0002-9904-1970-12521-6

On periodic solutions near homoclinic points

Michael Shub, Periodic orbits of hyperbolic diffeomorphisms and flows, Bull. Amer. Math. Soc. 75 (1969), 57–58. MR 234498, DOI 10.1090/S0002-9904-1969-12143-9
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
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
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
S. Smale, The $\Omega$-stability theorem, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 289–297. MR 0271971

Stability and genericity in dynamical systems

42

R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473–487. MR 217808, DOI 10.1016/0040-9383(67)90005-5
Robert F. Williams, 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 0235573
R. F. Williams, Zeta function in global analysis, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 335–339. MR 0266252
R. F. Williams, The $“\textrm {DA}''$ maps of Smale and structural stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 329–334. MR 0264705
Morris W. Hirsch, Foliations and noncompact transformation groups, Bull. Amer. Math. Soc. 76 (1970), 1020–1023. MR 292102, DOI 10.1090/S0002-9904-1970-12539-3
Carl P. Simon, On classification of a Baire set of diffeomorphisms, Bull. Amer. Math. Soc. 77 (1971), 783–787. MR 285018, DOI 10.1090/S0002-9904-1971-12806-9