  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.
