Periodic Points of Consecutive-Pattern-Avoiding Stack-Sorting Maps

West’s stack-sorting map involves a stack which avoids the permutation $21$ consecutively. Defant and Zheng extended this to a consecutive-pattern-avoiding stack-sorting map $S{C}_{\sigma }$, where the stack must always avoid a given permutation $\sigma$ consecutively. We address one of the main conjectures raised by Defant and Zheng in their dynamical approach to $S{C}_{\sigma }$. Specifically, we show that the periodic points of $S{C}_{\sigma }$ are precisely the permutations that consecutively avoid $\sigma$ and its reverse.