Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts

Details Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts

2007-09-17

arxiv.org/abs/0709.2667v2

Duke Mathematical Journal, 146, no. 2 (2009), 253-280

Mathematics

Dynamical Systems

Final version. To appear in Duke Mathematical Journal

Scientific paper

10.1215/00127094-2008-065

We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schr\"odinger operators, the obstruction vanishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schr\"odinger operator is a Cantor set.

Affiliated with

Also associated with

No associations

Rating

Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-379074