BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Zhenghe Zhang (UCR) DTSTART:20210429T170000Z DTEND:20210429T180000Z DTSTAMP:20260423T053136Z UID:Thouless/15 DESCRIPTION:Title: Positivity of the Lyapunov exponent for potentials generated by hyperbol ic transformations\nby Zhenghe Zhang (UCR) as part of UCI Mathematical Physics\n\n\nAbstract\nIn this talk\, I will introduce a recent work in s howing positivity of the Lyapunov exponent for Schr\\"odinger operators wi th potentials generated by hyperbolic dynamics. Specifically\, we showed t hat if the base dynamics is a subshift of finite type with an ergodic meas ure admitting a local product structure and if it has a fixed point\, then for all nonconstant H\\"older continuous potentials\, the set of energies with zero Lyapunov exponent is a discrete set. If the potentials are loca lly constant or globally fiber bunched\, then the set of zero Lyapunov exp onent is finite. We also showed that for generic such potentials\, we have full positivity in the general case and uniform postivity in the special cases. Such hyperbolic dynamics include expanding maps such as the doublin g map on the unit circle\, or Anosov diffeomorphism such as the Arnold's Cat map on 2-dimensional torus. It also can be applied to Markov chains w hose special cases include the i.i.d. random variable. This is a joint wit h A. Avila and D. Damanik.\n LOCATION:https://researchseminars.org/talk/Thouless/15/ END:VEVENT END:VCALENDAR