Cantor spectrum for schrödinger operators with potentials arising from generalized skew-shifts

Artur Avila, Jairo Bochi, David Damanik

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

We consider continuous SL(2,R{double-struck})-cocycles over a strictly ergodic homeomorphism that fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle that is not uniformly hyperbolic can be approximated by one that is conjugate to an SO(2,R{double-struck})-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 C0-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger 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 Schrödinger operator is a Cantor set.

Original languageEnglish (US)
Pages (from-to)253-280
Number of pages28
JournalDuke Mathematical Journal
Volume146
Issue number2
DOIs
StatePublished - Feb 2009

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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