Roots of iterates of maps

Let f : X → X be a selfmap of a path-connected space and let a ϵ X. We call x ϵ X an irreducible root of, fn, at a if fn(x) = a but fm(x) ≠ a for all m < n, where fn denotes the nth iterate of f. A lower bound N In(f;a) for the number of irreducible roots of fn is defined that is homotopy invariant under suitable hypotheses and, in many cases, can be calculated algebraically from the homomorphism of the fundamental group of X induced by f. In particular, this is true when X is a compact orientable manifold without boundary and f has nonzero degree. The geometric content of these calculations is described in concrete examples in terms of the discrete semidynamical system determined by f. We apply the theory of roots of iterates to primitive roots of unity in H-spaces.