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 the forth order are shown. From these explicit constructions, 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. (paper)