Recursive structure in the definitions of gauge-invariant variables for
  any order perturbations
  The construction of gauge-invariant variables for any order perturbations is
discussed. Explicit constructions of the gauge-invariant variables for
perturbations to 4th order are shown. From these explicit construction, the
recursive structure in the definitions of gauge-invariant variables for any
order perturbations is found. Through this recursive structure, the
correspondence with the fully non-linear exact perturbations is briefly
discussed.
