The action and index spectra of a Hamiltonian diffeomorphism and their behavior under iterations carry important information about the periodic orbits of the diffeomorphism. In a recent joint work with Ginzburg, we proved that for a certain sequence of iterations of a Hamiltonian diffeomorphism, the minimal action-index gap is bounded from above when the ambient manifold is closed and symplectically aspherical. This theorem implies the Conley conjecture asserting that such a diffeomorphism has simple periodic orbits of arbitrarily large period. The proof uses the facts, also established in the same work, that an isolated fixed point remains isolated for admissible iterations and that the local Floer homology groups for all such iterations are isomorphic to each other up to a shift of degree. The latter result can be viewed as a Hamiltonian version of the Shub-Sullivan theorem on the index of an isolated fixed point. In this talk we will outline the proof of the bounded-gap theorem and, time permitting, touch upon some recent developments towards its generalizations.