## Centralisers of $C^{\infty }$ diffeomorphisms

HTML articles powered by AMS MathViewer

- 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, - 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

*Singularities of differentiable mappings*, Proc. Liverpool Singularities Sympos., I, Lecture Notes in Math., vol. 192, Springer-Verlag, Berlin and New York, 1971, p. 6.

## 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