Growth and Integrability of Some Birational Maps in Dimension Three

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 c3∈Bir(P3) with projectivities that permute the fixed points of c3 and the points over which c3 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.