Nambu-Hamiltonian flows associated with discrete maps

TL;DR

This paper demonstrates that for invertible differentiable maps, a Nambu-Hamiltonian flow can be constructed with one variable acting as time, establishing a connection between discrete maps and continuous flows.

Contribution

It introduces a method to associate Nambu-Hamiltonian flows with invertible discrete maps, highlighting a novel map-flow correspondence.

Findings

Existence of Nambu-Hamiltonian flows for invertible maps

One variable can serve as a time parameter in the flow

Examples illustrating the map-flow relationship

Abstract

For a differentiable map $(x_1,x_2,..., x_n)\to (X_1,X_2,..., X_n)$ that has an inverse, we show that there exists a Nambu-Hamiltonian flow in which one of the initial value, say $x_n$, of the map plays the role of time variable while the others remain fixed. We present various examples which exhibit the map-flow correspondence.