1,131 research outputs found
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 Painlev\'e equations is studied. For a family of
rational varieties obtained by blow-ups at 8 points in general position in
, 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 . 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 preserves a one
parameter family of quadratic surfaces and that it can therefore be reduced to
the action on .Comment: 23 page
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.Comment: 24 pages, 2 figures, platex fil
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 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 th iterate of every
Painlev\'e equation in sakai's list is at most and therefore its
algebraic entropy is zero.Comment: 10 pages, pLatex fil
Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic
We prove that the general isolevel set of the ultra-discrete periodic Toda
lattice is isomorphic to the tropical Jacobian associated with the tropical
spectral curve. This result implies that the theta function solution obtained
in the authors' previous paper is the complete solution. We also propose a
method to solve the initial value problem.Comment: 14 page
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