Renormalization recursions for Schro¨dinger Green functions on several fractal lattices are solved explicitly in the sense in which one solves a system of difference equations. In other words, explicit orbits are constructed for the dynamical systems corresponding to the rational, planar maps that comprise the renormalization recursions. This construction is possible for each energy in the Cantor set of the infinite lattice spectrum. Thus if i and j are sites on opposite corners of the lattice the Green functions x=${\mathit{G}}_{\mathit{i}\mathit{i}}$ and y=${\mathit{G}}_{\mathit{i}\mathit{j}}$ are obtained as closed formulas as a function of lattice size L for fixed energy. Although scaling of y with L exhibits superlocalized behavior at almost all energies, there is an infinite set of energies at which ‖y‖∼${\mathit{L}}^{\mathrm{\beta }}$. When semi-infinite one-dimensional chains serving as current leads are attached to the terminal sites i and j, the Kubo-Greenwood conductance g is given by a simple formula in terms of x,y and the Green function u for the end of a semi-infinite chain. For energies at which ‖y‖∼${\mathit{L}}^{\mathrm{\beta }}$, it is found that g scales as in weak localization g∼${\mathit{L}}_{\mathit{c}}^{\mathrm{\beta }}$ with ${\mathrm{\beta }}_{\mathit{c}}$=-2‖β‖. For each lattice studied, including the Sierpin´ski k+1 simplex (k the Euclidean embedding dimension), there is an infinite set of energies at which ${\mathrm{\beta }}_{\mathit{c}}$=0 so that g∼const as L→∞. There are no mobility edges. Scaling exponents and energies such that ‖y‖∼${\mathit{L}}^{\mathrm{\beta }}$ are characterized in terms of fixed cycles in the dynamics of a map of the form a→A(a) related by conjugation to the original recursions.

• Received 28 August 1992

DOI:https://doi.org/10.1103/PhysRevB.47.7847