Algebraic entropy and the space of initial values for discrete dynamical systems

Abstract

A method to calculate the algebraic entropy of a mapping which can be lifted to an isomorphism of a suitable rational surfaces (the space of initial values) are presented. It is shown that the degree of the $n$th iterate of such a mapping is given by its action on the Picard group of the space of initial values. It is also shown that the degree of the $n$th iterate of every Painlevé equation in sakai's list is at most $O(n^2)$ and therefore its algebraic entropy is zero.

10 pages, pLatex file

References in corpus (6)

• 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
• A geometric approach to singularity confinement and algebraic entropy
• Algebraic Entropy of Birational Maps with Invariant Curves

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• Bi-rational maps in four dimensions with two invariants
• Space of initial conditions and geometry of two 4-dimensional discrete Painlevé equations
• Geometry of an elliptic-difference equation related to Q4
• Singularity confinement as an integrability criterion
• Algebraic entropy of an extended Hietarinta-Viallet equation
• Calculating the algebraic entropy of mappings with unconfined singularities
• Algebraic entropy computations for lattice equations: why initial value problems do matter
• Degree Complexity of a Family of Birational Maps
• Full-deautonomisation of a lattice equation
• Integrability of Discrete Equations Modulo a Prime
• Algebraic entropy for semi-discrete equations
• 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
• Full-parameter discrete Painlevé systems from non-translational Cremona isometries
• Degree growth of lattice equations defined on a 3x3 stencil
• Restoring discrete Painlevé equations from an E${}_8^{(1)}$-associated one
• Algebraic entropy for hex systems
• Algebraic entropy for systems of quad equations