The geometric form of automaton mappings, recurrent and $Z$-recurrent definition of sequences

For automaton mappings we present a method to construct geometric images, a method for complexity estimate by geometric forms, a method of $Z$-recurrent definition of sequences. A method for complexity estimate for finite sequences by recurrent and $Z$-recurrent numerical indicators is proposed. Numerical indicators of recurrent and $Z$-recurrent definitions of sequences are systematized into the spectrum of recurrent definitions with 5 levels of numerical indicators. The spectrum includes the order of a recurrent form, the numerical characteristics of various types of recurrent sequences, etc.