# Detecting and determining preserved measures and integrals of birational maps

## TL;DR

This paper introduces a symbolic algebra method using discrete Darboux polynomials to identify preserved measures and integrals in rational maps, including non-rational ones, demonstrated through various examples including discretizations of integrable systems.

## Contribution

The paper presents a novel symbolic algebra approach leveraging cofactors and Darboux polynomials to detect and compute preserved measures and integrals of rational maps.

## Key findings

- Most rational preserved integrals can be found with sufficient computational power.
- The method successfully detects preserved measures in various examples.
- Applicable to discretizations of integrable differential equations.

## Abstract

In this paper we use the method of discrete Darboux polynomials to calculate preserved measures and integrals of rational maps. The approach is based on the use of cofactors and Darboux polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, most, if not all, rational preserved integrals can be found (and even some non-rational ones). We show, in a number of examples, how it is possible to use this method to both determine and detect preserved measures and integrals of the considered rational maps. Many of the examples arise from the Kahan-Hirota-Kimura discretization of completely integrable systems of ordinary differential equations.