An infinite list of elements of $S$ can be viewed as a function from the set $\mathbb N$ of all natural numbers to $S$.
By the way, in the field of computer science, a kind of infinite list, called a circular list, is sometimes used. A circular list is a list in which there is no first and last element, and the sequence of elements is repeated infinitely.
My question is, is there any known mathematical counterpart of a circular list? In particular, what mathematical function does a circular list correspond to?
If $n$ is the list length, use $\Bbb Z/n\Bbb Z$, in which indices satisfy $x+n=x$. You can then move $k$ places to the right by computing $x+k$ modulo $n$.