Dynamics of the birational maps arising from $F_0$ and $dP_3$ quivers

TL;DR

This paper investigates the dynamics of maps derived from $F_0$ and $dP_3$ quivers, revealing their conjugation to globally periodic maps and analyzing their integrability and solutions.

Contribution

It provides explicit conjugations to globally periodic maps and links their integrability to the original quiver-derived maps.

Findings

Reduced symplectic maps are conjugate to globally periodic maps

Explicit solutions for the discrete dynamical systems are provided

The relationship between integrability and dynamics is clarified

Abstract

The dynamics of the maps associated to $F_0$ and $dP_3$ quivers is studied in detail. We show that the corresponding reduced symplectic maps are conjugate to globally periodic maps by providing explicit conjugations. The dynamics in $\Rb^N_+$ of the original maps is obtained by lifting the dynamics of these globally periodic maps and the solution of the discrete dynamical systems generated by each map is given. A better understanding of the dynamics is achieved by considering first integrals. The relationship between the complete integrability of the globally periodic maps and the dynamics of the original maps is explored.