, Volume 272, Issue 2, pp 397-442
Date: 03 Apr 2007

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

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Abstract

In this paper we consider the quasi-periodic Schrödinger 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 Schrödinger operator is an eigenvalue with an exponentially decaying eigenfunction. This generalizes [39] to the multi-frequency case (d > 1).

Communicated by G. Gallavotti
Research partially supported by STINT (Institutional Grant 2002-2052), The Royal Swedish Academy of Sciences and SVeFUM