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 $${{\,\textrm{c}\,}}_3\in {{\,\textrm{Bir}\,}}(\mathbb {P}^3)$$ c 3 ∈ Bir ( P 3 ) with projectivities that permute the fixed points of $${{\,\textrm{c}\,}}_3$$ c 3 and the points over which $${{\,\textrm{c}\,}}_3$$ c 3 performs a divisorial contraction. 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.

An exercise in experimental mathematics: calculation of the algebraic entropy of a map

“…The generating function of both sequences ( g B for the degrees of the B's, and g p for the p k 's) are then readily obtained. g B = P (s − 1) (s + 1) (s 6 + s 5 + s 4 + s 3 + s 2 + s + 1) (s 4 + 1) (s 8 + 1) (s 16 ]ocnmp[…”

“…We are interested in situations where this approach cannot be used. In particular, going beyond the "easy" two-dimensional case, makes the singularity analysis much more intricate, see for example [14,15,16,17].…”