Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices

The one-dimensional, discrete Schrödinger equation is studied when the potential is allowed to take on two values, $V_A$ and $V_B$, which are arranged according to a generalized Fibonacci sequence. The problem is reduced to a dynamical map for the traces of the transfer matrices which are given recursively by $M_{l+1\mathrm{}}$=$M_{l\mathrm{-}1}$ $M_l^n$, where n is a positive integer. A related class of sequences whose transfer matrices obey the recursion formula $M_{l+1\mathrm{}}$=$M_{l\mathrm{-}1}^n$ $M_l$ is also investigated.