Ergod. Th. & Dynam. Sys. (1987), 7, 463-479 Printed in Great Britain A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds STEPHAN PELIKAN AND EDWARD E. SLAMINKA Department of Mathematics, University of Cincinatti, Cincinnati, Ohio 45221, USA; Department of Mathematics, Auburn University, Auburn, Alabama, USA (Received 23 May 1985 and revised 14 March 1986) Abstract. The study of area preserving maps of manifolds has an extensive history in the theory of dynamical systems. One interest has been in the behaviour of such maps near an isolated fixed point. In 1974 Carl Simon proved the existence of an upper bound for the index of an isolated fixed point for C k area preserving diffeomorphisms of a surface. We extend his result to homeomorphisms of an orientable two manifold. The proof utilizes the notion of free modification, developed by Morton Brown, and enlarges the scope of the problem to the consideration of 'nice' measures, i.e. uniformly equivalent to Lebesgue measure on compact sets. By suitably modifying the homeomorphism and the measure, we obtain the following theorem. THEOREM. Let h : M 2 -» M 2 be an orientation preserving homeomorphism of a smooth orientable two manifold which preserves area. If p is an isolated fixed point of h, then the index of p is < +1. 0. Introduction What restrictions does the hypothesis that a homeomorphism be area-preserving place on the dynamics of the map? In this paper we present a result about the possible dynamics of such a map in the neighbourhood of an isolated fixed point: the index of the point must be less than 2. This result may be viewed as a generalization of Simon's theorem [6], which is concerned with C k diffeomorphisms. Our method of proof is necessarily completely different from his, and appears to be applicable in a wide variety of situations. The method involves a procedure by which an area preserving homeomorphism is modified to produce a new homeomorphism which has the same fixed points, which preserves an equivalent measure, and which has a special canonical form. This procedure has its roots in the Brouwer Translation Arc Lemma, and employs the idea of a free modification of a homeomorphism developed by Brown [3], [4], and by Schmitt [5]. It should be noted that, although we are primarily concerned with homeomorphisms, the techniques we use work equally well in the setting of C k diffeomorphisms.