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 c 3 ∈ Bir(P 3 ) with projectivities that permute the fixed points of c 3 and the points over which 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. CONTENTS 1. Introduction 1 2. The KHK discretisation of the Euler top 7 3. The Standard Cremona Transformation in dimension three 12 4. The Cremona-cubes group C 15 5. The spaces of initial values and the algebraic entropy 19 6. Covariant linear systems 26 7. Construction of the invariants 30 8. Conclusions 37 References 42