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Transactions of the American Mathematical Society
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
MathSciNet review: 0317356
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0317356-8
PII: S 0002-9947(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