TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 3, Pages 1005–1020 S 0002-9947(99)02320-X Article electronically published on May 20, 1999 DECOMPOSITION THEOREMS FOR GROUPS OF DIFFEOMORPHISMS IN THE SPHERE R. DE LA LLAVE AND R. OBAYA Abstract. We study the algebraic structure of several groups of differentiable diffeomorphisms in S n . We show that any given sufficiently smooth diffeomor- phism 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. Introduction The goal of this paper is to prove several decomposition theorems for diffeomor- phism groups. These theorems, roughly, state that any element of the group can be written as a finite product of elements lying in a smaller subgroup endowed with special properties, such as symmetry. In our case, the groups considered will be groups of differentiable diffeomor- phisms of the sphere, and the subgroups into which we factor them will be groups of diffeomorphisms that commute with reflections across a plane and which are essentially one dimensional. The factors into which we can decompose a given map will be slightly less differentiable than the original one. When we consider groups of C ∞ diffeomorphisms, the factors are also C ∞ . There are several motivations for the study of theorems of this type. For example, in [5] it is shown that a theorem of this type for the circle can be used to solve the inverse problem for scattering of geodesic fields in surfaces of genus one. This problem admits the following physical interpretation: The given diffeomorphism of a circle can be thought of as the transformation to be effected by a lens to be constructed; then, find the distribution of the refractive index that produces the desired effect. The problem is such that if it is solved for two diffeomorphisms, it is solved for their composition. Moreover, for symmetric mappings a simple construction works. Hence, the decomposition theorem shows that it can be solved for all mappings. For this application the loss of differentiability incurred in the factorization does not essentially affect the conclusions. We also note that these theorems are analogues of the usual factorization theo- rems in Lie algebras, and they could be useful in the problem of computing repre- sentations of diffeomorphism groups [1]. If we consider these theorems as infinite dimensional versions of factorization theorems for Lie groups, one first difficulty is that for diffeomorphism groups, the Received by the editors October 24, 1997. 1991 Mathematics Subject Classification. Primary 58D05, 57S25, 57S05. Key words and phrases. Decomposition theorems, diffeomorphism groups. c 1999 American Mathematical Society 1005 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use