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We study Schrodinger operators with a one-frequency analytic potential,focusing on the transition between the two distinct local regimescharacteristic respectively of large and small potentials. From the dynamicalpoint of view, the transition signals the emergence of nonuniformhyperbolicity, so the dependence of the Lyapunov exponent with respect toparameters plays a central role in the analysis. Though often ill-behaved byconventional measures, we show that the Lyapunov exponent is in fact remarkablyregular in a ``stratified sense'' which we define: the irregularity comes fromthe matching of nice (analytic or smooth) functions along sets with complicatedgeometry. This result allows us to stablish that the ``critical set'' for thetransition has at most codimension one, so for a typical potential the set ofcritical energies is at most countable, hence typically not seen by spectralmeasures. Key to our approach are two results about the dependence of theLyapunov exponent of one-frequency $\SL(2,\C)$ cocycles with respect toperturbations in the imaginary direction: on one hand there is a severe``quantization'' restriction, and on the other hand ``regularity'' of thedependence characterizes uniform hyperbolicity when the Lyapunov exponent ispositive. Our method is independent of arithmetic conditions on the frequency.

Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity