Iteration digraphs of a linear function

Abstract

An iteration digraph $G\left(n\right)$ generated by the function $f\left(x\right)modn$ is a digraph on the set of vertices $V=\left\{0,1,\dots ,n-1\right\}$ with the directed edge set $E=\left\{\left(v,f\left(v\right)\right)\mid v\in V\right\}$. Focusing specifically on the function $f\left(x\right)=10xmodn$, we consider the structure of these graphs as it relates to the factors of $n$. The cycle lengths and number of cycles are determined for various sets of integers including powers of 2 and multiples of 3.