Rational parking functions and $(m, n)$-invariant sets

An $(m, n)$-parking function can be characterized as function f:[n] → [m] such that the partition obtained by reordering the values of f fits inside a right triangle with legs of length m and n. Recent work by McCammond, Thomas, and Williams define an action of words in [m]ⁿ on Rⁿ. They show that rational parking functions are exactly the words that admit fixed points under that action. An $(m, n)$-invariant set is a set Δ ⊂ Z such that Δ + m ⊂ Δ and Δ + n ⊂ Δ. In this work we define an action of words in [m]ⁿ on $(m, n)$-invariant sets by removing the jth m-generator from Δ. We show this action also characterizes $(m, n)$-parking functions. Further we show that each $(m, n)$-invariant set is fixed by a unique monotone parking function. By relating the actions on Rᵐ and on $(m, n)$-invariant sets we prove that the set of all the points in Rᵐ that can be fixed by a parking function is a union of points fixed by monotone parking functions. In the case when (m, n) =1 we characterize the set of periodic points of the action defined on Rᵐ and show that the algorithm reversing the Pak-Stanley map proposed by Gorsky, Mazin, and Vazirani converges in a finite amount of steps.