Session D40 - Poster Session I.
POSTER session, Monday afternoon, March 12
Exhibit Hall, Washington State Convention Center
Discrete dynamical \nobreaksystems of Cremona maps in n variables are well studied in connection with solvable lattice models, e.g. by Maillard and others in search of symmetries of the Yang-Baxter equations. Here we give an explicit solution to the dynamics of a Cremona map associated with the Ashkin-Teller model. Starting from the matrix of Boltzmann weights w,\,x, and y, of the Ashkin-Teller model, \[ m\,=\,\left[ \beginarraycccc w & x & y & x \cr x & w & x & y \cr y & x & w & x \cr x & y & x & w \cr \endarray \right] \] Bellon and Maillard derive a dynamical system for the map I \circ J, with I a matrix inversion and J taking the reciprocal of each matrix entry. These recursions admit dilation, and there is an additional conserved quantity, resulting in a complete linearization of the map. We give an explicit solution of this dynamical system for w,\,x and y as functions of the number n of iterations.