GROUP THEORETIC REDUCTION OF LAPLACIAN DYNAMICAL PROBLEMS ON FRACTAL LATTICES

Discrete forms of the Schrodinger equation, the diffusion equation, th e linearized Landau-Ginzburg equation, and discrete models for vibrati ons and spin dynamics belong to a class of Laplacian-based finite diff erence models. Real-space renormalization of such models on finitely r amified regular fractals is known to give exact recursion relations. I t is shown that these recursions commute with Lie groups representing continuous symmetries of the discrete models. Each such symmetry reduc es the order of the renormalization recursions by one, resulting in a system of recursions with one fewer variable. Group trajectories are o btained from inverse images of fixed and invariant sets of the recursi ons. A subset of the Laplacian finite difference models can be mapped by change of boundary conditions and time dependence to a diffusion pr oblem with closed boundaries. In such cases conservation of mass simpl ifies the group flow and obtaining the groups becomes easier. To illus trate this, the renormalization recursions for Green functions on four standard examples are decoupled. The examples are (1) the linear chai n, (2) an anisotropic version of Dhar's 3-simplex, similar to a model dealt with by Hood and Southern, (3) the fourfold coordinated Sierpins ki lattice of Rammal and of Domany et al., and (4) a form of the Vicse k lattice. Prospects for applying the group theoretic method to more g eneral dynamical systems are discussed.