Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Diffeomorphisms without periodic points

Author: J. F. Plante
Journal: Proc. Amer. Math. Soc. 88 (1983), 716-718
MSC: Primary 58F20; Secondary 57R30, 57S99
MathSciNet review: 702306
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that a compact smooth manifold admits a selfdiffeomorphism without periodic points if and only if its Euler characteristic is zero. When the manifold has dimension $ \ne 3$ it is shown that such a diffeomorphism exists which is also volume preserving. The proof of this latter result uses a result of Gromov concerning the existence of nonsingular divergence-free vector fields, so an alternate proof of Gromov's result is sketched.

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

  • [1] F. Fuller, The existence of periodic points, Ann. of Math. (2) 57 (1953), 229-230. MR 0052764 (14:669f)
  • [2] M. Gromov, Convex integration of differential relations, Izv. Akad. Nauk 37 (1973), 329-343. (Russian) MR 0413206 (54:1323)
  • [3] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. MR 0182927 (32:409)
  • [4] M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214-227. MR 0209602 (35:499)
  • [5] R. Robinson, Generic properties of conservative systems, Amer. J. Math. 92 (1970), 562-603. MR 0273640 (42:8517)
  • [6] W. Thurston, Existence of codimension one foliations, Ann. of Math. (2) 104 (1976), 249-268. MR 0425985 (54:13934)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F20, 57R30, 57S99

Retrieve articles in all journals with MSC: 58F20, 57R30, 57S99

Additional Information

Keywords: Diffeomorphism, volume preserving, periodic point, vector field, divergence-free, homotopy
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society