On the group of real analytic diffeomorphisms of a compact real analytic manifold

In this paper we exhibit a Lie group structure on the group of real analytic diffeomorphisms of a compact real analytic manifold. Further, we show that a variant of the Kupka-Smale theorem holds for the underlying topology of the above mentioned Lie group. Introduction. The overall objective of this work is to prove an analytic version of the Kupka-Smale theorem: Given a smooth compact manifold, M, there exists a subset R C Diff '(M) which is a countable intersection of open dense subsets of Diff r(M) so that if f C R and p is a periodic point of f of order n (i.e., p EE Pn(f then (i) the eigenvalues of Tp f n lie off the unit circle, and (ii) the stable manifold, WS( p), and unstable manifold, WU( p), through p are injectively immersed; further, for p C Pn( f ) WS( p), and the unstable manifold, Wu(q), q C Pm(f), m arbitrary, intersect transversally; symbolically, WS(p) rhi Wu(q). The significance of this theorem to the study of the stability problem in differential equations has been discussed by Smale [16]; this theorem has been extended to retarded functional differential equations [6]. The proofs of these theorems known to me rely heavily on transversality theory which in turn relies heavily on the implicit function theorem for Frechet differentiable functions defined in open subsets of Banach spaces. While this does not create a major theoretical difficulty in proving an analogue of the Kupka-Smale theorem for Diff ?(M) = lim Diff r(M) because of well-known propositions concerning inverse r limits, we have found that for M real analytic the path for proving an analogue of the Kupka-Smale theorem is more circumlocutious. The first problem we encountered was to find an appropriate "natural" topology for Diff '(M); our second set of problems was the discovery of a notion of differentiability in the context of our chosen topology for which an implicit function theorem holds which would be adequate for transversality theory. The topology that we found "natural" turned out to be equivalent to the classical Co topology studied by Van Hove [10] and J. S. E. Silva [15]; that is, an inductive limit of spaces of complex analytic functions. Colombeau [3] in an unpublished Received by the editors January 13, 1981 and, in revised form, September 28, 1981. 1980 Mathematics Subject Classification. Primary 58F15, 58H05. (D1982 American Mathematical Society 0002-9947/81 /0000-0095/$07.00