  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.
