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Transactions of the American Mathematical Society

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Decomposition theorems for groups of diffeomorphisms in the sphere


Authors: R. de la Llave and R. Obaya
Journal: Trans. Amer. Math. Soc. 352 (2000), 1005-1020
MSC (1991): Primary 58D05, 57S25, 57S05
Published electronically: May 20, 1999
MathSciNet review: 1608297
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Abstract: We study the algebraic structure of several groups of differentiable diffeomorphisms in $\mathbf{S}^n$. We show that any given sufficiently smooth diffeomorphism can be written as the composition of a finite number of diffeomorphisms which are symmetric under reflection, essentially one-dimensional and about as differentiable as the given one.


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  • 1. Frédéric Bien, Global representations of the diffeomorphism group of the circle, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., vol. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 89–107. MR 1026949
  • 2. Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222. MR 656198, 10.1090/S0273-0979-1982-15004-2
  • 3. Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR 0448362
  • 4. Steven G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Exposition. Math. 1 (1983), no. 3, 193–260. MR 782608
  • 5. Joel Langer and David A. Singer, Diffeomorphisms of the circle and geodesic fields on Riemann surfaces of genus one, Invent. Math. 69 (1982), no. 2, 229–242. MR 674403, 10.1007/BF01399503
  • 6. R. de la Llave, Remarks on J. Langer and D. A. Singer decomposition theorem for diffeomorphisms of the circle, Comm. Math. Phys. 104 (1986), no. 3, 387–401. MR 840743
  • 7. R. de la Llave, R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynamical Systems, 5 (1999), 157-184. Preprint available from http://www.ma.utexas.edu/mp_arc
  • 8. Jürgen Moser, A rapidly convergent iteration method and non-linear differential equations. II, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 499–535. MR 0206461
  • 9. Helmut Rüssmann, On optimal estimates for the solutions of linear difference equations on the circle, Proceedings of the Fifth Conference on Mathematical Methods in Celestial Mechanics (Oberwolfach, 1975), Part I, 1976, pp. 33–37. MR 0445973
  • 10. E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math. 28 (1975), 91–140. MR 0380867
  • 11. E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. II, Comm. Pure Appl. Math. 29 (1976), no. 1, 49–111. MR 0426055

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Additional Information

R. de la Llave
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: llave@math.utexas.edu

R. Obaya
Affiliation: Departamento Matemática Aplicada a la Ingeniería, Escuela Superior de Ingenieros Industriales, Universidad de Valladolid, 47011 Valladolid, Spain
Email: rafoba@wmatem.eis.uva.es

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02320-X
Keywords: Decomposition theorems, diffeomorphism groups
Received by editor(s): October 24, 1997
Published electronically: May 20, 1999
Article copyright: © Copyright 1999 American Mathematical Society