Fixed Points of Multiple-valued Transformations

A multiple-valued transformation T from a space X to a space Y is a function assigning to each point x o f X a nonempty closed subset T(x) of F. The graph of T comprises those points (x, y) in the topological product X Y for which y belongs to T(x). All spaces to be considered shall be compact metric and all transformations T shall be upper semi-continuous, meaning that their graphs are closed, hence compact, subsets of X Y. When the domain X and the range Y of T coincide, a fixed point of T is defined to be a point x which belongs to its image set T(x). The fixed points correspond to those points in the product XX which belong to the intersection of the graph of T with the diagonal of XX. In the special case that X is an orientable w-manifold the diagonal carries an w-cycle D. If, in addition, each neighborhood of the graph contains a representative cycle of an w-cycle homology class T whose intersection number with D is not zero, then the diagonal must meet the graph of T, so that under the assumptions made at least one fixed point must exist. To each w-cycle class T having representative cycles in each neighborhood of the graph corresponds an endomorphism T*T of the homology group H(X) of X, determined as follows: starting with any £-cycle 7, form in the product XX the upright cylinder y XX, intersect the cylinder with T and project the intersection laterally into X to obtain finally T*(y). If the homology of X uses a field as coefficient group, then the endomorphisms !T*r, constitute a vector space *(T). For a single-valued transformation r, the space *(r) comprises scalar multiples of the conventional endomorphism r* induced by r. The definition of *(T), here described for manifolds only, has been extended to an arbitrary A.N.R., using singular homology, by Lefschetz [s] and to an arbitrary compact metric space, using Cech homology, by O'Neill [ l l ] . The Lefschetz fixed point theorem, extended to multiple-valued transformations, assumes the following form: Let X be an A.N.R. and let T be an upper semi-continuous multiple-valued transformation of X into itself. Then either T has a fixed point or else the equa-