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

An $(m, n)$-parking function can be characterized as function $f:[n] \to [m]$ 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 $\mathbb{R}^n$. 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 $\Delta \subset \mathbb{Z}$ such that $\Delta + m \subset \Delta$ and $\Delta + n \subset \Delta$. In this work we define an action of words in $[m]^n$ on $(m, n)$-invariant sets by removing the $j$th $m$-generator from $\Delta$. 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 $\mathbb{R}^m$ and on $(m, n)$-invariant sets we prove that the set of all the points in $\mathbb{R}^m$ that can be fixed by a parking function is a union of points fixed by monotone parking functions. In the case when $\gcd(m, n) =1$ we characterize the set of periodic points of the action defined on $\mathbb{R}^m$ and show that the algorithm reversing the Pak-Stanley map proposed by Gorsky, Mazin, and Vazirani converges in a finite amount of steps.