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Continuity of Lyapunov Exponents and Cantor spectrum for a class of $C^2$ Quasiperiodic Schr\"odinger Cocycles

Date: 
Wednesday 8th April 2015 - 15:00 to 16:00
Venue: 
INI Seminar Room 1
Abstract: 
Co-author: Zhenghe Zhang (Rice University)

We show that for a class of $C^2$ quasiperiodic potentials and for any fixed \emph{Diophantine} frequency, the Lyapunov exponents of the corresponding Schr\"odinger cocycles are uniformly positive and weakly H\"older continuous as function of energies. Moreover, we show that the spectrum is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general quasiperiodic $\mathrm{SL}(2,\R)$ cocycles.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons