Schrodinger operators with potentials generated by hyperbolic transformations: I-positivity of the Lyapunov exponent

• 成果类型: Article
• 署名作者: Avila, Artur; Damanik, David; Zhang, Zhenghe
• 署名单位: University of Zurich; Instituto Nacional de Matematica Pura e Aplicada (IMPA); Rice University; University of California System; University of California Riverside
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01157-2
• 发表日期: 2023
• 页码: 851-927
• 关键词: deterministic products; anderson localization; cocycles; continuity; systems; THEOREM; bounds

摘要

We consider discrete one-dimensional Schrodinger 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 Holder 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 Holder 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 Holder continuous potentials. In particular, we apply our results to Schrodinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains.