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Abstract We consider dynamically defined Hermitian matrices generated from orbits of the doubling map. We prove that their spectra fall into the GUE universality class from random matrix theory. 

We study the one-dimensional discrete Schr\"odinger operator with the skew-shift potential $$2\lambda \cos\2π(j^2\omega+jy+x)$$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $$\lambda>0$$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $$L(\lambda)$$ at small $$\lambda$$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $$L(\lambda)$$ is fully consistent with $$L(\lambda)$$ being positive and satisfying the usual Figotin-Pastur type asymptotics $$L(\lambda)\sim C\lambda^2$$ as $$\lambda\to 0$$. The analogous quantity behaves completely differently in the Almost-Mathieu model, whose zero-energy Lyapunov exponent vanishes for $$\lambda<1$$. The main technical work consists in establishing good lower bounds on the exponential sums (quadratic Weyl sums) that appear in our perturbation series.