Centralisers of $C^{\infty }$ diffeomorphisms
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- by Boyd Anderson PDF
- Trans. Amer. Math. Soc. 222 (1976), 97-106 Request permission
Abstract:
It is shown that if F is a hyperbolic contraction of ${R^n}$, coordinates may be chosen so that not only is F a polynomial mapping, but so is any diffeomorphism which commutes with F. This implies an identity principle for diffeomorphisms ${G_1}$ and ${G_2}$ commuting with an arbitrary Morse-Smale diffeomorphism F of a compact manifold M: if ${G_1},{G_2} \in Z(F)$, then ${G_1} = {G_2}$ on an open subset of $M \Rightarrow {G_1} \equiv {G_2}$ on M. Finally it is shown that under a certain linearisability condition at the saddles of F, $Z(F)$ is in fact a Lie group in its induced topology.References
- George D. Birkhoff, Dynamical systems, American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966. With an addendum by Jurgen Moser. MR 0209095
- Kuo-Tsai Chen, Local diffeomorphisms—$C^{\infty }$ realization of formal properities, Amer. J. Math. 87 (1965), 140–157. MR 173271, DOI 10.2307/2373228
- Nancy Kopell, Commuting diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 165–184. MR 0270396 H. I. Levine, Singularities of differentiable mappings, Proc. Liverpool Singularities Sympos., I, Lecture Notes in Math., vol. 192, Springer-Verlag, Berlin and New York, 1971, p. 6.
- John N. Mather, Stability of $C^{\infty }$ mappings. II. Infinitesimal stability implies stability, Ann. of Math. (2) 89 (1969), 254–291. MR 259953, DOI 10.2307/1970668
- 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
- Shlomo Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809–824. MR 96853, DOI 10.2307/2372437
- Shlomo Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. II, Amer. J. Math. 80 (1958), 623–631. MR 96854, DOI 10.2307/2372774
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 222 (1976), 97-106
- MSC: Primary 58F99; Secondary 57D50
- DOI: https://doi.org/10.1090/S0002-9947-1976-0423424-6
- MathSciNet review: 0423424