Iteration of Functions $f:X^{k}\rightarrow X$ and their Periodicity

Abstract

We propose a notion of iterating functions $f:X^{k}\rightarrow X$ in a way that represents recurrence relations of the form $a_{n+k}=f(a_{n},a_{n+1},...,a_{n+k-1})$. We define a function as $n$-involutory when its $n$th iterate is the identity map, and discuss elementary group-theoretic properties of such functions along with their relation to cycles of their corresponding recurrence relations. Further, it is shown that a function $f:X^{k}\rightarrow X$ that is 2-involutory in each of its $k$ arguments (holding others fixed) is $(k+1)$-involutory.