Group theoretic reduction of Laplacian dynamical problems on fractal lattices

Discrete forms of the Schr\"odinger equation, the diffusion equation, the linearized Landau-Ginzburg equation, and discrete models for vibrations and spin dynamics belong to a class of Laplacian-based finite difference models. Real-space renormalization of such models on finitely ramified regular fractals is known to give exact recursion relations. It is shown that these recursions commute with Lie groups representing continuous symmetries of the discrete models. Each such symmetry reduces the order of the renormalization recursions by one, resulting in a system of recursions with one fewer variable. Group trajectories are obtained from inverse images of fixed and invariant sets of the recursions. A subset of the Laplacian finite difference models can be mapped by change of boundary conditions and time dependence to a diffusion problem with closed boundaries. In such cases conservation of mass simplifies the group flow and obtaining the groups becomes easier. To illustrate this, the renormalization recursions for Green functions on four standard examples are decoupled. The examples are (1) the linear chain, (2) an anisotropic version of Dhar's 3-simplex, similar to a model dealt with by Hood and Southern, (3) the fourfold coordinated Sierpin-acuteski lattice of Rammal and of Domany et al., and (4) a form of the Vicsek lattice. Prospects for applying the group theoretic method to more general dynamical systems are discussed.