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 . 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.

**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**

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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:
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