Lyapunov Positivity and Monodromy Criteria for Moving-Law Random--Quasiperiodic Cocycles
We study moving-law random-environment matrix cocycles whose one-step quasiperiodic cocycle law varies stationarily along an ergodic environment orbit. Two fixed-law structures then cease to suffice: a single stationary projective measure does not record transport between successive environment fibers, and pooling the moving supports may create products that are not dynamically admissible along any environment orbit. We replace the first by environment-indexed stationary projective families and prove an environment-wise Furstenberg formula. When the top Lyapunov exponent vanishes, averaged stationarity upgrades to exact equivariance at the support level, \(g_*η^ω=η^{Θω}\), which forces either an environment-wise compact reduction or a finite support-invariant family of nonzero proper measurable subbundles. Environment-wise noncompactness and environment-wise strong irreducibility therefore imply positivity. To address the second obstruction, we compare admissible products along matched environment segments with equal total torus translation. Thus their quotients are genuine returns on a common vector fiber and generate environment-matched monodromy groups. Stable noncompactness and stable strong irreducibility of these groups verify the cocycle-level hypotheses. Applied to stationarily varying random--quasiperiodic Schrödinger cocycles, the framework yields strict positivity of the top Lyapunov exponent at every real energy under the stated assumptions.