Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

Let M be a smooth compact and simply-connected manifold with simply-connected boundary (partial M), r be a fixed odd natural number. We consider f, a (C^1) self-map of M, preserving (partial M). Under the assumption that the dimension of M is at least 4, we define an invariant (D_r(f;M,partial M)) that is equal to the minimal number of r-periodic points for all maps preserving (partial M) and (C^1)-homotopic to f. As an application, we give necessary and sufficient conditions for a reduction of a set of r-periodic points to one point in the (C^1)-homotopy class.