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Invariant volume form for 3D QRT maps

Jaume Alonso, Yuri B. Suris

TL;DR

This work proves the existence of an invariant volume form for 3D QRT maps, defined by two pencils of quadrics in ${\mathbb P}^3$, thereby establishing their Liouville-type integrability in odd dimensions. The authors deploy pencil-adapted coordinates to reduce the 3D dynamics to 2D QRT maps on quadric intersections, derive a density $\rho(x)=Q_{\infty}(X)P_{\infty}(X)$ that yields an invariant measure, and prove this measure is chart-independent. Central to the argument is a key proposition linking the Jacobian $\det\frac{\partial(x_1,x_2,x_3)}{\partial(x,y,\lambda)}$ to $-Q_{\infty}(X)$, which they verify via two illustrative examples and a general homogeneous-coordinate calculation. The results unify the treatment of 3D QRT maps with discrete Painlevé theory and Kahan discretizations, suggesting broad implications for discrete integrability and future applications.

Abstract

Recently, we proposed a three-dimensional generalization of QRT maps. These novel maps can be associated with pairs of pencils of quadrics in $\mathbb P^3$. By construction, these maps have two rational integrals (parameters of both pencils). In the present paper, we find an invariant volume form for these maps, thus finally establishing their integrability.

Invariant volume form for 3D QRT maps

TL;DR

This work proves the existence of an invariant volume form for 3D QRT maps, defined by two pencils of quadrics in , thereby establishing their Liouville-type integrability in odd dimensions. The authors deploy pencil-adapted coordinates to reduce the 3D dynamics to 2D QRT maps on quadric intersections, derive a density that yields an invariant measure, and prove this measure is chart-independent. Central to the argument is a key proposition linking the Jacobian to , which they verify via two illustrative examples and a general homogeneous-coordinate calculation. The results unify the treatment of 3D QRT maps with discrete Painlevé theory and Kahan discretizations, suggesting broad implications for discrete integrability and future applications.

Abstract

Recently, we proposed a three-dimensional generalization of QRT maps. These novel maps can be associated with pairs of pencils of quadrics in . By construction, these maps have two rational integrals (parameters of both pencils). In the present paper, we find an invariant volume form for these maps, thus finally establishing their integrability.

Paper Structure

This paper contains 10 sections, 4 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. 2D QRT maps
  3. 3D QRT maps
  4. Pencil-adapted coordinates
  5. Main theorem
  6. Proof of Theorem \ref{['theorem main']}
  7. Examples
  8. Proof of Proposition \ref{['prop Q infty']}
  9. Conclusions
  10. Appendix: Proof of the invariant area form for 2D QRT maps

Key Result

Proposition 1

The map $f$ has an invariant area form $dx\wedge dy/C_{\infty}(x,y)$ with the density $C_{\infty}(x,y)$. This can be expressed as where we denote $(\widetilde{x},\widetilde{y})=f(x,y)$.

Theorems & Definitions (6)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof