Growth and integrability of some birational maps in dimension three

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.

arXiv comment48 pages, 6 figures, 7 tables, comments are welcome