Finite-step projective dynamics of Schrödinger cocycles: non-degeneracy, regularity, and rigidity

Abstract

We study the no deterministic images condition (NDI) for Schrödinger cocycles, which excludes a common image of a fixed probability measure under almost all projective transfer transformations. We prove two-step NDI at every energy for nonconstant measurable potentials over mixing systems and ergodic potentials with infinite essential range, and classify persistence for finite-state locally constant potentials. For bounded Markov and hidden Markov potentials on general state spaces under a one-step $L^2$ norm gap condition on the Markov operator, we prove Hölder bounds for solution-direction distributions above an exponential cutoff and Hölder conditional stationary measures. Counterexamples show the essential sharpness of both assumptions. For nonconstant real trigonometric potentials, including the almost Mathieu operator, we prove NDI and Hölder bounds for distributions of solution directions at all steps $k\ge2$ and all frequencies (rational or irrational). The exponent depends on the step, while the constant is uniform in steps, frequency and energy on compact intervals. For every irrational frequency and every energy with positive Lyapunov exponent, we prove non-atomicity of both Oseledets direction distributions and convergence to the unstable distribution from every phase-independent initial probability. This gives a new criterion for a zero Lyapunov exponent: a subsequential limit with an atom for one fixed initial direction suffices. All these results extend to an analytic class, with step-dependent regularity bounds.