Algebraic Entropy for lattice equations

Abstract

We give the basic definition of algebraic entropy for lattice equations. The entropy is a canonical measure of the complexity of the dynamics they define. Its vanishing is a signal of integrability, and can be used as a powerful integrability detector. It is also conjectured to take remarkable values (algebraic integers).

References in corpus (5)

• Algebraic entropy
• Singularity confinement and chaos in discrete systems
• Integrable lattice equations and their growth properties
• Complexity spectrum of some discrete dynamical systems
• On the complexity of some birational transformations

Cited by in corpus (19)

• Weak Lax pairs for lattice equations
• On the integrability of a new lattice equation found by multiple scale analysis
• Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations I: First integrals
• Algebraic entropy for differential-delay equations
• Integrable Lattice Maps: $Q_5$, a Rational Version of $Q_4$
• Linearizability and fake Lax pair for a consistent around the cube nonlinear non-autonomous quad-graph equation
• Full-deautonomisation of a lattice equation
• Classification of quad-equations on a cuboctahedron
• Search for CAC-integrable homogeneous quadratic triplets of quad equations and their classification by BT and Lax
• Algebraic entropy for face-centered quad equations
• A systematic approach to reductions of type-Q ABS equations
• An algebraic approach to discrete time integrability
• Discrete nonlinear hyperbolic equations. Classification of integrable cases
• Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice
• Algebraic entropy for hex systems
• Algebraic entropy for systems of quad equations
• A non autonomous generalization of the $Q_{\text{V}}$ equation
• A completeness study on a class of discrete, 'two by two' Lax pairs
• Linear degree growth in lattice equations