Abstract
AbstractMotivated 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$${{\,\textrm{c}\,}}_3\in {{\,\textrm{Bir}\,}}(\mathbb {P}^3)$$c3∈Bir(P3)with projectivities that permute the fixed points of$${{\,\textrm{c}\,}}_3$$c3and the points over which$${{\,\textrm{c}\,}}_3$$c3performs 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.
Funder
Gruppo Nazionale per la Fisica Matematica
Ministero dell’Università e della Ricerca
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Nuclear and High Energy Physics,Statistical and Nonlinear Physics
Reference96 articles.
1. Ablowitz, M.J., Halburd, R., Herbst, B.: On the extension of the Painlevé property to difference equations. Nonlinearity 13, 889–905 (2000)
2. Alonso, J., Suris, Y. B., Wei, K.: A three-dimensional generalization of QRT maps. 2022. arXiv:2207.06051 [nlin.SI]
3. Anglès d’Auriac, J.-C., Maillard, J.-M., Viallet, C.M.: A classification of four-state spin edge Potts models. J. Phys. A: Math. Gen. 35, 9251–9272 (2002)
4. Arnol’d, V.I.: Dynamics of complexity of intersections. Bol. Soc. Bras. Mat. 21, 1–10 (1990)
5. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60, 2nd edn. Springer, Berlin (1997)