# Fixed Points of Parking Functions

## TL;DR

This paper introduces a new characterization of rational parking functions via fixed points of an action, and proves the invertibility of the associated zeta map, connecting it to existing combinatorial structures.

## Contribution

It provides a novel fixed point perspective on rational parking functions and establishes the invertibility of the zeta map in this context.

## Key findings

- Characterization of rational parking functions through fixed points.
- Proof of invertibility of the zeta map for coprime parameters.
- Connection of the zeta map to the sweep map on rational Dyck paths.

## Abstract

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.