In this paper, we study the behavior of the local Floer homology
of an isolated fixed point and the growth of the action gap under
iterations. We prove that an isolated fixed point of a diffeomorphism
remains isolated for the so-called admissible iterations and that the
local Floer homology groups of a Hamiltonian diffeomorphism for such
iterations are isomorphic to each other up to a shift of degree. Furthermore,
we study the pair-of-pants product in local Floer homology, and
characterize a particular class of isolated fixed points (the symplectically
degenerate maxima), which plays an important role in the proof
of the Conley conjecture. Finally, we apply these results to show that
for a quasi-arithmetic sequence of admissible iterations of a Hamiltonian
diffeomorphism with isolated fixed points the minimal action gap
is bounded from above when the ambient manifold is closed and symplectically
aspherical. This theorem is a generalization of the Conley
conjecture.