arXiv subjects

Tomoyuki Takenawa

Publications and source records attributed to Tomoyuki Takenawa.

26 records · Page 2Linked to original sources

arXiv2008

Tropical spectral curves and integrable cellular automata

We propose a method to study the integrable cellular automata with periodic boundary conditions, via the tropical spectral curve and its Jacobian. We introduce the tropical version of eigenvector map from the isolevel set to a divisor class on the tropical hyperelliptic curve. We also provide some conjectures related to the divisor class and the Jacobian. Finally, we apply our method to the periodic box and ball system and clarify the algebro-geometrical meaning of the real torus introduced for its initial value problem.

math-ph↗

arXiv2007

Integrability of $n$-dimensional dynamical systems of type $E_7^{(1)}$ and $E_8^{(1)}$

We propose an $n$-dimensional analogue of elliptic difference Painlevé equation. Some Weyl group acts on a family of rational varieties obtained by successive blow-ups at $m$ points in $\mpp^n(\mc)$, and in many cases they include the affine Weyl groups with symmetric Cartan matrices as subgroups. It is shown that the dynamical systems obtained by translations of these affine Weyl groups possess commuting flows and that their degrees grow quadratically. For the $E_7^{(1)}$ and $E_8^{(1)}$ cases, existence of preserved quantities is investigated. The elliptic difference case is also studied.

nlin.SI↗

arXiv2005

Elliptic curves and birational representation of Weyl groups

Some Weyl group acts on a family of rational varieties obtained by successive blow-ups at $m$ ($m\geq n+2$) points in the projective space $\mpp^n(\mc)$. In this paper we study the case where all the points of blow-ups lie on a certain elliptic curve in $\mpp^n$. Investigating the action of Weyl group on the Picard groups on the elliptic curve and on rational varieties, we show that the action on the parameters can be written as a group of linear transformations on the $(m+1)$-st power of a torus.

math.AG↗

arXiv2003

Discrete dynamical systems associated with the configuration space of 8 points in P^3(C)

A 3 dimensional analogue of Sakai's theory concerning the relation between rational surfaces and discrete Painlevé equations is studied. For a family of rational varieties obtained by blow-ups at 8 points in general position in ${\mathbb P}^3$, we define its symmetry group using the inner product that is associated with the intersection numbers and show that the group is isomorphic to the Weyl group of type $E_7^{(1)}$. By normalizing the configuration space by means of elliptic curves, the action of the Weyl group and the dynamical system associated with a translation are explicitly described. As a result, it is found that the action of the Weyl group on ${\mathbb P}^3$ preserves a one parameter family of quadratic surfaces and that it can therefore be reduced to the action on ${\mathbb P}^1\times {\mathbb P}^1$.

nlin.SI↗

arXiv2002

The extended Weyl group $\widetilde{W}(D_5^{(1)})$ as an extension of KNY's birational representation of $\widetilde{W}(A_1^{(1)}\times A_3^{(1)})$

We study the birational representation of $\wt{W}(A_1^{(1)}\times A_3^{(1)})$ proposed by Kajiwara-Noumi-Yamada (KNY) in the case of $m=2$ and $n=4$. It is shown that the equation can be lifted to an automorphism of a family of $A_3^{(1)}$ surfaces and therefore the group of Cremona isometries is $\wt{W}(D_5^{(1)})$ ($\supset \wt{W}(A_1^{(1)}\times A_3^{(1)})$). The equation can be decomposed into two mappings which are conjugate to the $q$-$P_{VI}$ equation. It is also shown that the subgroup of Cremona isometries which commute with the original translation is isomorphic to $\mz \times \wt{W}(A_3^{(1)}) \times \wt{W}(A_1^{(1)})$.

nlin.SI↗

arXiv2001

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 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.

nlin.SI↗

arXiv2001

Discrete dynamical systems associated with root systems of indefinite type

A geometric charactrization of the equation found by Hietarinta and Viallet, which satisfies the singularity confinement criterion but which 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 a realization of a Cremona isometry on the Picard group of the surface. It is also shown that the group of Cremona isometries is isomorphic to an extended Weyl group of indefinite type. A method to construct the mappings associated with some root systems of indefinite type is also presented.

nlin.SI↗

arXiv2001

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 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.

nlin.SI↗