  The relationship between mappings of sets and renormalization group
transformations is established, and renormalization group invariants of such
mappings are found. These results are valid both for continuous and discrete
mappings and for various dimensionality of image and preimage of the mappings
too. It is suggested a number of mapping approximations improved in comparison
with an ordinary power expansion. The approximations take into account the
global one-to-one character of the mappings. The method is illustrated by a
number of examples: by reconstructing of some analytical functions, calculating
the integral of the typical partition function of statistical mechanics and the
ground state energy for the quartic anharmonic oscillator. In the whole range
of nonlinearity parameter varying from zero up to infinity the accuracy of the
RG approximation based on a few terms of divergent series is about 0.06% in the
next to last case and 0.004% in the last one.
