Growth and integrability of some birational maps in dimension three

TL;DR

This paper investigates the growth and integrability of certain birational maps in three dimensions, revealing three distinct behaviors related to their algebraic complexity and invariants, with implications for algebraic geometry and dynamical systems.

Contribution

It characterizes the growth and integrability of specific Cremona group elements in three dimensions using algebraic geometry techniques, identifying three distinct dynamical behaviors.

Findings

Maps can be integrable with quadratic degree growth and two invariants.

Some maps are periodic with two-periodic degree sequences and more than two invariants.

Other maps are non-integrable with submaximal degree growth and one invariant.

Abstract

Motivated by the study of the Kahan--Hirota--Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation $\mathrm{c}_3\in\mathrm{Bir}(\mathbb{P}^3)$ with projectivities that permute the fixed points of $\mathrm{c}_3$ and the points over which $\mathrm{c}_3$ performs a divisorial contraction. More specifically, we show that three behaviour are possible: (A) integrable with quadratic degree growth and two invariants, (B) periodic with two-periodic degree sequences and more than two invariants, and (C) non-integrable with submaximal degree growth and one invariant.