A geometric approach to singularity confinement and algebraic entropy

Abstract

A geometric approach to the equation found by Hietarinta and Viallet, which satisfies the singularity confinement criterion but exhibits chaotic behavior, is presented. It is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be the action of an extended Weyl group of indefinite type. A method to calculate its algebraic entropy by using the theory of intersection numbers is presented.

9 pages, 1 figure, LaTeX file

References in corpus (4)

• Algebraic entropy
• Singularity confinement and chaos in discrete systems
• A $q$-anaolg of the sixth Painlevé equation
• Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations

Cited by in corpus (14)

• Discrete integrable systems and Poisson algebras from cluster maps
• Algebraic entropy and the space of initial values for discrete dynamical systems
• On the complexity of some birational transformations
• On the algebraic structure of rational discrete dynamical systems
• Third-order integrable difference equations generated by a pair of second-order equations
• Singularity confinement as an integrability criterion
• Singularity confinement in delay-differential Painlevé equations
• Algebraic entropy of an extended Hietarinta-Viallet equation
• On the singularity structure of the discrete KdV equation
• Degree Complexity of a Family of Birational Maps
• A systematic method for constructing discrete Painlevé equations in the degeneration cascade of the E${}_8$ group
• Degree growth of lattice equations defined on a 3x3 stencil
• Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice
• Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property