Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent

Abstract We consider discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if function is non-constant Hölder continuous defined on subshift finite type with fully supported ergodic measure admitting local product structure and fixed point, then Lyapunov exponent positive away from set energies. Moreover, for functions in residual subset space functions, everywhere. If locally constant or globally fiber bunched Lyapuonv set. an open dense question, uniformly positive. Our results can be applied to any measures equilibrium states potentials. In particular, apply our over expanding maps unit circle, automorphisms finite-dimensional torus, Markov chains.