Let's consider one dimensional cellular automaton. It is build upon its rule, i.e. a function $f : S^3 \rightarrow S$, where $S = \{0,1\}$. The case described is the elementary cellular automaton, because the rule-function has 3 bits wide input.
The automaton transforms its infinite rows using the rule. Let's limit the rows to pseudo-infinite ones, i.e. left end of row is connected to right end, and the row length is $N$, a finite number.
The question: each row has its unique successor – the result of multiple productions of $f$. This defines a new function. Let's denote it as $F$. The function takes whole row as input, and outputs whole new row. Are such functions $F$ – i.e. based on basic functions like $f$ – investigated in some mathematics topic? I am interested in properties of such functions.