Over the past decade integrability for rational maps (but also for lattice equations) has come to be defined through a « zero algebraic entropy » requirement, and a lot of research effort has gone into the development of novel rigorous methods to calculate the entropy for a given mapping or equation. For mappings on the complex (projective) plane this definition fits very well with a more analytic notion of « integrability », in terms of the existence or non-existence of conserved quantities. In higher dimensions, or for lattice equations, such a link is far from clear and one could argue that the zero algebraic entropy requirement is perhaps too coarse as a meaningful integrability criterion and that a more refined definition is needed.

In the first part of the talk I will give an overview of the different notions involved in detecting integrability in rational mappings, in particular algebraic entropy and singularity confinement, and I will present a novel and quite spectacular link between both notions for mappings of the plane. In the latter part of the talk I will comment on the paucity of results for mappings in higher dimensions and I will discuss several problems with the entropy approach to defining integrability in the higher dimensional (or lattice) case.

This talk is partly based on

• A. Stokes, T. Mase, R. Willox and B. Grammaticos, Deautonomisation by singularity confinement and degree growth, The Journal of Geometric Analysis 35, 65 (2025)
  and
• A Ramani, B Grammaticos, A S Carstea and R Willox, Obtaining the growth of higher order mapping through the study of singularities, Journal of Physics A: Mathematical and Theoretical 58(11) 115201 (2025)