The Existence of Periodic Points

Abstract

A periodic point of a transformation of a set into itself is a point which is carried back to its original position by some iterate of the transformation. The purpose of this paper is to demonstrate topological conditions which ensure the existence of periodic points. THEOREM 1. Let T be a mapping of an orientable combinatorial manifold M into itself, and suppose that the corresponding homomorphism T* of rational p-cycle classes is an isomorphism onto. Let u be a rational p-cycle class and let v be any other rational cycle class of M. Let M be given a definite orientation. Then if the intersection cycle classes (T u, v), (T'u, v), , (T u, v) are all zero, where B is the p-dimensional Betti number of M, it follows that the intersection cycle class (u, v) is zero. PROOF. The rational p-cycle classes of M form a vector space of dimension B. Hence the B + 1 cycle classes u, T*u, ... , T*u must be related by a nontrivial dependency c0u + cjT*u + *. + CBT*U = 0. Let Ck, be the first nonzero coefficient. By hypothesis, T* has an inverse. Operating on the dependency by 71k gives CkU + Ck+lT*u + * + cBT* u = 0. Intersecting this combination of cycle classes with v and dividing by Ck gives the conclusion (u, v) = 0. THEOREM 2. Let T be a homeomorphism of an orientable combinatorial manifold M onto itself. If the Euler characteristic of M is not zero, then T has a periodic point. PROOF. Theorem 2 is a consequence of Theorem 1 applied to the orientable combinatorial product manifold M X M. Let M be given a definite orientation, thereby inducing an orientation of M X M. Let D be the cycle class in M X Ml which is the image of the basic cycle on M under the diagonal mapping x -* (x, x) of M into M X M. The homeomorphism T determines a homeomorphism 1 X T: (x, y) -+ (x, Ty) of M X M onto itself. If T has no periodic point, then the intersection classes ((1 X T)* D, D), ((1 X T) D, D), * a must all vanish. Hence by Theorem 1 the intersection class (D, D) must vanish. But a known result asserts that the algebraic value of the 0-cycle class (D, D) is equal within sign to the Euler characteristic of M. Since the Euler characteristic of M was assumed to be different from zero, T must have a periodic point. If Theorems 1 and 2 are established for manifolds with boundary then Theorem 2 can be extended to complexes by the imbedding method used by Lefschetz to extend his fixed-point formula. It is simpler, however, to make direct use of the Lefschetz formula. The method below was used by J. Nielsen in an investigation of transformations of surfaces. THEOREM 3. Let T be a mapping of a complex K into itself which induces auto-