Theory of quasiperiodic lattices. II. Generic trace map and invariant surface

Abstract

The generic trace map is established for binary quasiperiodic lattices represented by any irrational number. This is a generalization of the renormalization-group method of Kohmoto, Kadanoff, and Tang (KKT) for the Fibonacci lattice. All the trace maps for any quasiperiodic lattice preserve the same invariant surface first discovered by KKT. There is a special set of points on the invariant surface, which we call the invariant six-cycle. These are fixed points of the trace maps. We are able to obtain representations of the trace maps or scaling transformations on the invariant six-cycle. This enables us to determine the periods of the trace maps, which are very important in order to know the scaling property of the electronic wave function and energy spectrum at the band center.