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

Abstract

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.