Relative periodic point theory

In this paper we give methods for computing lower bounds on the number of periodic points of a self-map of a topological pair (X,Y). These methods are particularly useful if X⧹Y is disconnected, as we obtain a symbolic description of orbits in terms of components of X⧹Y. The theories are homotopy invariant, and also allow the comparison of maps on different spaces. In many cases, we can also obtain a lower bound for topological entropy.