  We prove the transversality result necessary for defining local Morse chain
complexes with finite cyclic group symmetry. Our arguments use special
regularized distance functions constructed using classical covering lemmas, and
an inductive perturbation process indexed by the strata of the isotropy set. A
global existence theorem for symmetric Morse-Smale pairs is also proved.
Regarding applications, we focus on Hamiltonian dynamics and rigorously
establish a local contact homology package based on discrete action
functionals. We prove a persistence theorem, analogous to the classical
shifting lemma for geodesics, asserting that the iteration map is an
isomorphism for good and admissible iterations. We also consider a
Chas-Sullivan product on non-invariant local Morse homology, which plays the
role of pair-of-pants product, and study its relationship to symplectically
degenerate maxima. Finally, we explore how our invariants can be used to study
bifurcation of critical points (and periodic points) under additional
symmetries.
