IMPORTANT NOTICE

The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at cust-serv@ams.org or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).

 

Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-99-02320-X
Published electronically: May 20, 1999
MathSciNet review: 1608297
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


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

  • 1. F. Bien, Global representations of the diffeomorphism group of the circle, Infinite dimensional Lie algebras and groups, Marseille 1988, World Scientific, 1989, pp. 89-107. MR 90j:22020
  • 2. R. Hamilton, The inverse function theorem of Nash and Moser, Bull. A.M.S., 7 (1982), 65-222. MR 83j:58014
  • 3. M. Hirsch, Differential Topology, Springer-Verlag, 1976. MR 56:6669
  • 4. S. Krantz, Lipschitz spaces, smoothness of functions and approximation theory, Exposition. Math. 3 (1983), 193-260. MR 86g:41001
  • 5. J. Langer, D.A. Singer, Diffeomorphisms of the circle and geodesic fields on Riemann surfaces of genus one, Invent. Mat. 69 (1982), 229-242. MR 84h:58115
  • 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), 387-401. MR 87h:58174
  • 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. J. Moser, A rapidly convergent iteration method and non-linear differential equations II, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 499-535. MR 34:6280
  • 9. H. Rüssmann, On optimal estimates for the solutions of linear difference equations on the circle, Celestial Mech., 14 (1976), 33-37. MR 56:4306
  • 10. E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems I, Comm. Pure and Appl. Math., 28 (1975), 91-140. MR 52:1764
  • 11. E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems II, Comm. Pure and Appl. Math., 29 (1976), 49-111. MR 54:14001

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58D05, 57S25, 57S05

Retrieve articles in all journals with MSC (1991): 58D05, 57S25, 57S05


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: https://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

American Mathematical Society