Avila, ArturDamanik, DavidZhang, Zhenghe2023-02-232023-02-232023Avila, Artur, Damanik, David and Zhang, Zhenghe. "Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent." <i>Inventiones mathematicae,</i> 231, (2023) Springer Nature: 851-927. https://doi.org/10.1007/s00222-022-01157-2.https://hdl.handle.net/1911/114479We 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 the sampling function is a non-constant Hölder continuous function defined on a subshift of finite type with a fully supported ergodic measure admitting a local product structure and a fixed point, then the Lyapunov exponent is positive away from a discrete set of energies. Moreover, for sampling functions in a residual subset of the space of Hölder continuous functions, the Lyapunov exponent is positive everywhere. If we consider locally constant or globally fiber bunched sampling functions, then the Lyapuonv exponent is positive away from a finite set. Moreover, for sampling functions in an open and dense subset of the space in question, the Lyapunov exponent is uniformly positive. Our results can be applied to any subshift of finite type with ergodic measures that are equilibrium states of Hölder continuous potentials. In particular, we apply our results to Schrödinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains.engThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.Journal articles00222-022-01157-2https://doi.org/10.1007/s00222-022-01157-2