Dynamics of the Quasi-Periodic Schrödinger Cocycle at the Lowest Energy in the Spectrum

Abstract

In this paper we consider the quasi-periodic Schrodinger cocycle over \(\mathbb{T}^d\) (d ≥ 1) and, in particular, its projectivization. In the regime of large coupling constants and Diophantine frequencies, we give an affirmative answer to a question posed by M. Herman [21, p.482] concerning the geometric structure of certain Strange Non-chaotic Attractors which appear in the projective dynamical system. We also show that for some phase, the lowest energy in the spectrum of the associated Schrodinger operator is an eigenvalue with an exponentially decaying eigenfunction. This generalizes [39] to the multi-frequency case (d > 1).