
A natural variational problem on a finite type von Neuman algebra is D  > Det(D+m)=exp(tr log(D+m)), where m is a complex parameter and the functional is restricted to some manifold of operators for which the determinant exists. The first nontrivial contribution of this functional is tr(D^4) which becomes essentially the Wilson functional in lattice gauge theory if the variations are restricted to classes of Laplacians. It is a discrete version of the MaxwellYangMills functional. Critical points lead to nonlinear wave equations. "Journal of Physics A, 29, L595L600, 1996" Using a variational trick of Aubry one can show that the corresponding coupled map lattice has infinitedimensional horseshoes "A. Jacobson and O. Knill, Journal of Physics A, 29, L595L600, 1996". Already primitive experiments in one dimension show interesting dynamics. In the simplest possible situation, where $D$ is invariant under space translations, the time evolution is a Henon type symplectic map. 
