Tomoyuki Takenawa

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

A method to calculate the algebraic entropy of a mapping, which can be lifted to an isomorphism of a suitable rational surface (the space of initial values), is presented. It is shown that the degree of the nth iterate of such a mapping is given by its action on the Picard group of the space of initial values. It is also shown by construction that the degree of the nth iterate of every Painleve equation in Sakais list is O(n2) and therefore its algebraic entropy is zero.

A geometric approach to singularity confinement and algebraic entropy

A geometric approach to the equation found by Hietarinta and Viallet, which satisfies the singularity confinement criterion but exhibits chaotic behaviour, 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.

A Tropical Analogue of Fay's Trisecant Identity and the Ultra-Discrete Periodic Toda Lattice

We introduce a tropical analogue of Fay’s trisecant identity for a special family of hyperelliptic tropical curves. We apply it to obtain the general solution of the ultra-discrete Toda lattice with periodic boundary conditions in terms of the tropical Riemann’s theta function.

A note on minimization of rational surfaces obtained from birational dynamical systems

In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable mappings, including discrete versions of reduced Nahm equations.

A classification of two-dimensional integrable mappings and rational elliptic surfaces

We classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces. While the QRT mappings can be restricted on each fiber, there exist several classes of integrable mappings which exchange fibers. We also show an equivalent condition when a generalized Halphen surface becomes a Halphen surface of index m.

On the identity of two q -discrete Painlevé equations and their geometrical derivation

We show that two recently discovered q-discrete Painlevé equations are one and the same system. Moreover we provide a novel derivation of this q-discrete system based on transformations obtained with the help of affine Weyl groups.

Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel–Roberts–Thompson mappings, can be deautonomized to discrete Painleve equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painleve equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painleve equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painleve equations, including examples whose symmetry groups do not appear explicitly in Sakais classification.

On the (Non-)Differentiability of the Optimal Value Function When the Optimal Solution is Unique

We present an example of a parameterized optimization problem, with a continuous objective function differentiable with respect to the parameter, that admits a unique optimal solution, but whose optimal value function is not differentiable. We also show independence of Danskins and Milgrom and Segals envelope theorems.