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

TL;DR

This paper introduces a new way to iterate functions of multiple variables to model recurrence relations, exploring their algebraic properties and involutory characteristics.

Contribution

It defines a novel iteration method for multivariable functions and analyzes their involutory properties and relation to recurrence cycles.

Findings

Functions that are 2-involutory in each argument are (k+1)-involutory.

The paper establishes group-theoretic properties of these functions.

Connections between involutory functions and recurrence relation cycles are discussed.

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.