# Re-factorising a QRT map

## TL;DR

This paper investigates the decomposition of QRT maps into involutions, revealing new classes of maps and connections to known integrable systems like HKY and Yang-Baxter maps.

## Contribution

It provides new criteria and methods for decomposing QRT maps into involutions, expanding the understanding of their structure and related integrable maps.

## Key findings

- Identified conditions for decomposing QRT maps into involutions
- Discovered new classes of maps arising from these decompositions
- Connected decompositions to known HKY and Yang-Baxter maps

## Abstract

A QRT map is the composition of two involutions on a biquadratic curve: one switching the $x$-coordinates of two intersection points with a given horizontal line, and the other switching the $y$-coordinates of two intersections with a vertical line. Given a QRT map, a natural question is to ask whether it allows a decomposition into further involutions. Here we provide new answers to this question and show how they lead to a new class of maps, as well as known HKY maps and quadrirational Yang-Baxter maps.