Let A = nj°e£ 0 gΓfc(Λ) be a complete graded (associative or Lie) algebra over a field of characteristic zero, filtered by the decreasing filtration Fj(A) = Π^L, & k (A). We let Aut(Λ) denote the group of filtration preserving automorphisms of A, and Aut o (Λ) the subgroup consisting of those elements of AuX(A) which preserve the grading. In this paper we prove that every element of Aut(A) has a unique polar decomposition of the form u 0 exp(d), where u 0 e Aut o (A) and d :A -> A is a

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... n increasing derivation. 219 220 A. BAIDER AND R. C. CHURCHILL presentation here is that the relationships between the group structures are formulated explicitly. The splittings of §5 and §6 are only the first step toward "normal form" theory. How such forms can be achieved by transforming coordinates is well-known (for vector fields and diffeomorphisms see e.g. Takens [12] and [13], and for Hamiltonian systems see e.g. Moser [9]), and will not be addressed here, although we briefly touch on the subject in examples ending these sections. For detailed historical surveys of normal form theory see van der Meer [14, Brjuno [3" and Dixon and Esterle [5, pp. 152-3]. The authors would like to thank the referee for constructive comments and criticisms regarding the original manuscript, and Richard Cushman for useful discussions of the literature. Jo ' J Jo 7 = 1,..., j 0 . To any graded group (vector space, algebra) X = Πy> 7o Xj we associate the filtration Fj(X) = Tl n >jX n , and to any filtered group (vector 222 A.