  We present the theory of non-stationary normal forms for uniformly
contracting smooth extensions with sufficiently narrow Mather spectrum. We give
coherent proofs of existence, (non)uniqueness, and a description of the
centralizer results. As a corollary, we obtain corresponding results for normal
forms along an invariant contracting foliation. The main improvements over the
previous results in the narrow spectrum setting include explicit description of
non-uniqueness and obtaining results in any regularity above the precise
critical level, which is especially useful for the centralizer. In addition to
sub-resonance normal form, we also prove corresponding results for resonance
normal form, which is new in the narrow spectrum setting.
