Diffeomorphisms without periodic points
HTML articles powered by AMS MathViewer
- by J. F. Plante PDF
- Proc. Amer. Math. Soc. 88 (1983), 716-718 Request permission
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
- F. B. Fuller, The existence of periodic points, Ann. of Math. (2) 57 (1953), 229–230. MR 52764, DOI 10.2307/1969856
- M. L. Gromov, Convex integration of differential relations. I, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 329–343 (Russian). MR 0413206
- Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 182927, DOI 10.1090/S0002-9947-1965-0182927-5
- 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
- R. Clark Robinson, Generic properties of conservative systems, Amer. J. Math. 92 (1970), 562–603. MR 273640, DOI 10.2307/2373361
- W. P. Thurston, Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268. MR 425985, DOI 10.2307/1971047
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 88 (1983), 716-718
- MSC: Primary 58F20; Secondary 57R30, 57S99
- DOI: https://doi.org/10.1090/S0002-9939-1983-0702306-5
- MathSciNet review: 702306