Critical states and fractal attractors. in fractal tongues: Localization in the Harper map - art. no. 045204

Localized states of Harper's equation correspond to strange nonchaotic attr actors in the related Harper mapping, In parameter space, these fractal att ractors with nonpositive Lyapunov exponents occur in fractally organized to ngue-like regions which emanate from the Cantor set of eigenvalues. on the critical line epsilon =1. A topological invariant characterizes wave functi ons corresponding to energies in the gaps in the spectrum. This permits a u nique integer labeling of the gaps and also determines their scaling proper ties as a function of, potential strength.