Algebraic entropy

Abstract

For any discrete time dynamical system with a rational evolution, we define an entropy, which is a global index of complexity for the evolution map. We analyze its basic properties and its relations to the singularities and the irreversibility of the map. We indicate how it can be exactly calculated.

Latex, 12 pages, 1 figure

Cited by in corpus (127)

• Continuous Symmetries of Difference Equations
• Geometric Aspects of Painlevé Equations
• Integrable lattice equations and their growth properties
• Discrete integrable systems and Poisson algebras from cluster maps
• Recursion operators, conservation laws and integrability conditions for difference equations
• Diophantine Integrability
• Cluster mutation-periodic quivers and associated Laurent sequences
• Algebraic entropy and the space of initial values for discrete dynamical systems
• Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations
• The Generalized Symmetry Method for Discrete Equations
• Algebraic Entropy for lattice equations
• Discrete dynamical systems associated with root systems of indefinite type
• Searching for integrable lattice maps using factorization
• Integrability properties of Kahan's method
• A geometric approach to singularity confinement and algebraic entropy
• Linearisable Mappings and the Low-Growth Criterion
• On the complexity of some birational transformations
• Integrable systems without the Painlevé property
• Algebraic Entropy of Birational Maps with Invariant Curves
• Affine linear and D4 symmetric lattice equations : symmetry analysis and reductions
• Method for searching higher symmetries for quad graph equations
• On the algebraic structure of rational discrete dynamical systems
• Symplectic Maps from Cluster Algebras
• Discretization of polynomial vector fields by polarization
• Solvable Chaos
• A classification of four-state spin edge Potts models
• Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type
• Darboux Integrability of Trapezoidal $H^4$ and $H^6$ Families of Lattice Equations II: General Solutions
• Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations I: First integrals
• Singularity confinement and full-deautonomisation: a discrete integrability criterion
• Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras
• Calculating algebraic entropies: an express method
• Singularity confinement and algebraic integrability
• Bi-rational maps in four dimensions with two invariants
• Space of initial conditions and geometry of two 4-dimensional discrete Painlevé equations
• Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups
• Geometry of an elliptic-difference equation related to Q4
• Algebraic entropy for differential-delay equations
• Rational Maps with Invariant Surfaces
• Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations
• Singularity confinement for matrix discrete Painleve Equations
• Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space
• Irreducibility and co-primeness as an integrability criterion for discrete equations
• Integrable Lattice Maps: $Q_5$, a Rational Version of $Q_4$
• On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties
• Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency
• Algebraic entropy of an extended Hietarinta-Viallet equation
• Singularity confinement and chaos in two-dimensional discrete systems
• Calculating the algebraic entropy of mappings with unconfined singularities
• On some classes of discrete polynomials and ordinary difference equations
• Invariant varieties of periodic points for some higher dimensional integrable maps
• Algebraic entropy computations for lattice equations: why initial value problems do matter
• Classification of integrable discrete Klein-Gordon models
• Integrable lattice equations with vertex and bond variables
• Integrability criterion in terms of coprime property for the discrete Toda equation
• Coalgebra symmetry for discrete systems
• The ${\mathfrak{sl}}_2(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems
• Degree Complexity of a Family of Birational Maps
• Integrability of Discrete Equations Modulo a Prime
• Full-deautonomisation of a lattice equation
• Lie group analysis of a generalized Krichever-Novikov differential-difference equation
• Multi-indexed Wilson and Askey-Wilson Polynomials
• Algebraic entropy for semi-discrete equations
• The noncommutative geometry of elliptic difference equations
• An algebraic method of classification of S-integrable discrete models
• Algebraic entropy for face-centered quad equations
• Algebraic entropy of a class of five-point differential-difference equations
• Height growth of solutions and a discrete Painlevé equation
• The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
• Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation
• An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps
• An exercise in experimental mathematics: calculation of the algebraic entropy of a map
• From Polygons to Ultradiscrete Painlevé Equations
• A systematic approach to reductions of type-Q ABS equations
• On the Coprimeness Property of Discrete Systems without the Irreducibility Condition
• From integrable equations to Laurent recurrences
• The solution to the initial value problem for the ultradiscrete Somos-4 and 5 equations
• Full-parameter discrete Painlevé systems from non-translational Cremona isometries
• Algebraical Entropy and Arrow of Time
• Birational Mappings and Matrix Sub-algebra from the Chiral Potts Model
• Nonlinear forms of coprimeness preserving extensions to the Somos-$4$ recurrence and the two-dimensional Toda lattice equation --investigation into their extended Laurent properties--
• Algebraic entropy for algebraic maps
• Degree growth of lattice equations defined on a 3x3 stencil
• Algebraic entropy of sign-stable mutation loops
• A characterization of pseudo-Anosov mapping classes via tropical cluster transformations
• Studies on spaces of initial conditions for nonautonomous mappings of the plane
• Reconstructing a Lattice Equation: a Non-Autonomous Approach to the Hietarinta Equation
• Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice
• Generators of rank 2 cluster algebras of affine types via linearization of seed mutations
• Dynamics of Non-polar Solutions to the Discrete Painlevé I Equation
• Growth of heights in piecewise-affine planar maps
• A non autonomous generalization of the $Q_{\text{V}}$ equation
• Algebraic entropy for systems of quad equations
• Degree-growth of monomial maps
• Algebraic entropy for hex systems
• How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6
• Studies on the discrete integrable equations over finite fields
• Linear Recurrences in the Degree Sequences of Monomial Mappings
• Bi-orthogonal systems on the unit circle, regular semi-classical weights and the discrete Garnier equations
• From auto-Bäcklund transformations to auto-Bäcklund transformations, and torqued ABS equations
• A family of integrable and non-integrable difference equations arising from cluster algebras
• Algebraic entropy of a multi-term recurrence of the Hietarinta-Viallet type
• A two dimensional lattice equation as an extension of the Heideman-Hogan recurrence
• Three approaches to detecting discrete integrability
• Quantum integrability: Lagrangian 1-form case
• QRT maps and related Laurent systems
• The dynamical degree of billiards in an algebraic curve
• Discrete integrable equations on face-centered cubics: consistency and Lax pairs of corner equations
• Entropy of holomorphic and rational maps: a survey
• Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices
• Degree complexity of birational maps related to matrix inversion: Symmetric case
• Symmetries in Connection Preserving Deformations
• Finite-order meromorphic solutions and the discrete Painleve equations
• Boundary Qoutinet Graphs and Graph Index
• Singularity confinement and proliferation of tau functions for a general differential-difference Sawada-Kotera equation
• Integrability of one degree of freedom symplectic maps with polar singularities
• On the Degree Growth of Birational Mappings in Higher Dimension
• Kahan-Hirota-Kimura maps preserving original cubic hamiltonians
• Dynamics of a two parameter family of plane birational maps: maximal entropy
• Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property
• Toda type equations over multi-dimensional lattices
• A lemma on the difference quotients
• On the degree growth of iterated birational maps
• Reduction of polynomial dynamical systems modulo primes
• The nature of manifolds of periodic points for higher dimensional integrable maps
• Degree Complexity of Matrix Inversion
• Periodicity, linearizability and integrability in seed mutations of type $A_N^{(1)}$