Quasi-periodic Operators Research Articles

Nonlinear dynamics of localization in a class of one-dimensional quasicrystals.

We consider the diagonalization of quasiperiodic operators on generalizations of the Fibonacci lattice defined recursively by $(A,B)\ensuremath{\rightarrow}({A}^{n}B,A)$. The inflation symmetry of these lattices induces a three-dimensional nonlinear dynamical map on the traces of associated transfer matrices. We find the invariant manifolds for these trace maps to be twisted and pinched versions of the Fibonacci manifold. We investigate the effect of these pinches on the spectrum of a tight-binding Hamiltonian and consider the limit of weak incommensurability: $n\ensuremath{\rightarrow}\ensuremath{\infty}$.