Attractors of Parikh mapping iterations

Three types of the Parikh mapping are introduced, namely, alphabetic, alphabetic-basis and basis. Explicit expressions for attractors of the k-th order in bases n >= 8, including countable ones, are found. Properties for the alphabetic, alphabetic-basis and basis Parikh vectors are given at each step of the Parikh mapping. The maximum number of iterations leading to attractors of the k-th order in the basis n is determined.