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The \textit{order of appearance} $ z(n) $ of a positive integer $ n $ in theFibonacci sequence is defined as the smallest positive integer $ j $ such that$ n $ divides the $ j $-th Fibonacci number. A \textit{fixed point} ariseswhen, 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 theform $ 5^{k} $ or $ 12\cdot5^{k} $ for all non-negative integers $ k $. Itimmediately follows that there are infinitely many fixed points in theFibonacci sequence. We prove that there are infinitely many integers thatiterate to a fixed point in exactly $ k $ steps. In addition, we constructinfinite families of integers that go to each fixed point of the form $12 \cdot5^{k}$. We conclude by providing an alternate proof that all positive integers$n$ reach a fixed point after a finite number of iterations.

Dynamics of the Fibonacci Order of Appearance Map