I will introduce some recent work regarding positivity and large deviations of the Lyapunov exponent for Schrodinger operators with potentials generated by hyperbolic dynamics. We show that for certain special classes of potentials, the Lyapunov exponent is positive away from a finite set. Moreover, a uniform large deviation estimate holds true away from an arbitrary small neighborhood of this finite set. With these two results, we obtain a full spectral Anderson localization for the corresponding operators almost surely. This is a joint work with A. Avila and D. Damanik.

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