$(H,\rho)$-induced dynamics and the quantum game of life

We propose an extended version of quantum dynamics for a certain system S, whose evolution is ruled by a Hamiltonian $H$, its initial conditions, and a suitable set $\rho$ of {\em rules}, acting repeatedly on S. The resulting dynamics is not necessarily periodic or quasi-periodic, as one could imagine for conservative systems with a finite number of degrees of freedom. In fact, it may have quite different behaviors depending on the explicit forms of $H$, $\rho$ as well as on the initial conditions. After a general discussion on this $(H,\rho)$-{\em induced dynamics}, we apply our general ideas to extend the classical game of life, and we analyze several aspects of this extension.