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.

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

“…can be lifted to an automorphism of a rational surfaces X obtained by successive 14 blow-ups from P 1 × P 1 [6]. Hence its Picard group is…”

“…where a i ∈ C and a 1 , a 3 , a 6 are nonzero and an over-line means the value of the image by the mapping [6]. In this case the coefficients of H i and E i do not change and therefore the degrees and the algebraic entropy do not change, since its action on the Picard group is identical with the action of the original autonomous version.…”

“…A sequence of rational surfaces X i is (or X i themselves are) called the space of initial values for the sequence of ϕ i if each ϕ i is lifted to an isomorphism, i.e. bi-holomorphic mapping, from X i to X i+1 [5,6,7]. Here, the mapping ϕ ′ is called a mapping lifted from the mapping ϕ if ϕ ′ coincides with ϕ on any point where ϕ is defined.…”

“…We apply our method to the mapping which was found by Hietarinta and Viallet [1] and whose space of initial values are obtained by 14 blow-ups from P 1 × P 1 [6]. We simplify calculation using the root systems associated with the symmetries of their space of initial values.…”

Discrete Dynamical Systems¶Associated with Root Systems of Indefinite Type

“…H 0 , H 1 , E 9 , E 10 E 11 , E 12 , E 13 , E 14 → H 0 + α 3 , H 1 + α 3 , E 9 + α 3 , E 10 + α 3 E 11 + α 3,1 , E 12 + α 3,2 , E 13 + α 3,2 , E 14 + α 3,1 (16) We define the action of σ 12 and σ 13 on Pic(X) as follows.…”