Growth and integrability of some birational maps in dimension three

2023/01/22 by Michele Graffeo, Giorgio Gubbiotti, Graffeo, Michele +1 · 1 citation
Mathematics · #14E15 #39A36 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Primary 14E07 #Secondary 14H70

paper · pdf · doi:10.48550/arxiv.2301.09129

openalex publication_date 2023/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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

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