Algebraic entropy for differential-delay equations

We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations

On the algebraic structure of rational discrete dynamical systems

We show how singularities shape the evolution of rational discrete dynamical systems. The stabilisation of the form of the iterates suggests a description providing among other things generalised Hirota form, exact evaluation of the algebraic entropy as well as remarkable polynomial factorisation properties. We illustrate the phenomenon explicitly with examples covering a wide range of models

Scattering of cosmic strings by black holes: loop formation

We study the deformation of a long cosmic string by a nearby rotating black hole. We examine whether the deformation of a cosmic string, induced by the gravitational field of a Kerr black hole, may lead to the formation of a loop of cosmic string. The segment of the string which enters the ergosphere of a rotating black hole gets deformed and, if it is sufficiently twisted, it can self-intersect chopping off a loop of cosmic string. We find that the formation of a loop, via this mechanism, is a rare event. It will only arise in a small region of the collision phase space, which depends on the string velocity, the impact parameter and the black hole angular momentum. We conclude that generically, the cosmic string is simply scattered or captured by the rotating black hole.Comment: 11 pages, 2 figures, RevTe

Integrable lattice equations with vertex and bond variables

We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as non-autonomous "Yang-Baxter maps". We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with algebraic entropy as well as multidimensional consistencyComment: 15 pages, remarks added, other minor change

How to detect the integrability of discrete systems

International audienceSeveral integrability tests for discrete equations will be reviewed. All tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as Lax pairs. Specifically, singularity confinement, algebraic entropy, Nevanlinna theory, Diophantine integrability and discrete systems over finite fields will be described