Dynamics of the Fibonacci Order of Appearance Map

The \textit{order of appearance} $z(n)$ of a positive integer $n$ in the Fibonacci sequence is defined as the smallest positive integer $j$ such that $n$ divides the $j$-th Fibonacci number. A \textit{fixed point} arises when, for a positive integer $n$, we have that the $n^{\text{th}}$ Fibonacci number is the smallest Fibonacci that $n$ divides. In other words, $z(n) = n$. In 2012, Marques proved that fixed points occur only when $n$ is of the form $5^{k}$ or $12\cdot5^{k}$ for all non-negative integers $k$. It immediately follows that there are infinitely many fixed points in the Fibonacci sequence. We prove that there are infinitely many integers that iterate to a fixed point in exactly $k$ steps. In addition, we construct infinite families of integers that go to each fixed point of the form $12 \cdot 5^{k}$. We conclude by providing an alternate proof that all positive integers $n$ reach a fixed point after a finite number of iterations.