论文标题
Schrödinger操作员具有由双曲线转换产生的电势：I。Lyapunov指数的阳性
Schrödinger Operators With Potentials Generated by Hyperbolic Transformations: I. Positivity of the Lyapunov Exponent
论文作者
论文摘要
我们考虑了离散的一维Schrödinger运算符，其电势是通过沿一般双曲线转换的轨道采样而产生的。具体而言，我们表明，如果采样函数是在有限类型上定义的非恒定Hölder连续函数，则具有沿局部产品结构和固定点的厄尔冈量度，那么Lyapunov指数将远离离散的能量集。此外，对于在Hölder连续函数的残留子集中采样函数，Lyapunov指数无处不在。如果我们考虑局部恒定或全球纤维堆的采样函数，则Lyapuonv指数远离有限集。此外，对于所讨论空间的开放和致密子集中的采样函数，Lyapunov指数均匀地为正。我们的结果可以应用于有限类型的任何子缩影，并采用具有hölder持续电势的均衡状态的千古措施。特别是，我们将结果应用于Schrödinger操作员，这些操作员在单位圆的扩展图，有限维圆环的双曲线自动形态和马尔可夫链上定义。
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 the sampling function is a non-constant Hölder continuous function defined on a subshift of finite type with an 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.
