Fixed points of parking functions

We define an action of words in \$[m]^n\$ on \$\mathbb{R}^m\$ to give a new characterization of rational parking functions—they are exactly those words whose action has a fixed point. We use this viewpoint to give a simple definition of Gorsky, Mazin, and Vazirani's zeta map on rational parking functions when \$m\$ and \$n\$ are coprime, and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington's sweep map on rational Dyck paths.  This is joint work with Jon McCammond and Hugh Thomas.