Let $A_1$ be the finite set of periodic points in the Julia set for a rational map, $R$, which has lowest possible period.
Inductively, define $A_{n+1} = R^{-1} (A_n) \cup A_n$.
Why is it true that $A_n$ is finite for all $n$?
For context, I am looking at Sullivan's proof to the No Wandering Domain Theorem (https://www.math.stonybrook.edu/~bishop/classes/math627.S13/Sullivan-1985-Nonwandering.pdf) on page 410.
Any help would be greatly appreciated.