Nielsen coincidence theory of (n,m)-valued pairs of maps

We consider pairs of maps (f, g), where f is an n-valued map and g is an m-valued map, defined on connected finite polyhedra. A point x such that f(x)∩ g(x)≠ ∅ is called a coincidence point of f and g. A useful device for studying coincidence points would be a Nielsen-type invariant which provides a lower bound for the number of coincidence points of all (n, m)-valued pairs of maps homotopic to (f, g). The construction of such an invariant N(f : g) was proposed in [J. Fixed Point Theory Appl. 14, 309–324 (2013)]. Building on this approach, we observe that the definition of the coincidence set used there requires a correction in order to properly account for coincidences arising as intersections of the graphs of the two maps. We therefore introduce a graph-based formulation of coincidence classes and define the corresponding geometric and algebraic Nielsen coincidence numbers. In the case of (n, m)-valued pairs of maps of the circle our invariant provides a sharp lower bound, which we precisely determine.