Parametric Furstenberg Theorem on random products of SL(2, R) matrices
Parametric Furstenberg Theorem on random products of SL(2, R) matrices
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DOI：
10.1016/j.aim.2020.107522
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发表时间：
2021-02-12
影响因子：
1.7
通讯作者：
Kleptsyn, Victor
中科院分区：
文献类型：
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作者：
Gorodetski, Anton;Kleptsyn, Victor
我们考虑SL（2，R）矩阵的随机产品，取决于一个参数在一个非一致双曲制度。我们表明，如果对参数的依赖是单调的，那么几乎可以肯定的随机产品具有上（limsup）李雅普诺夫指数是等于规定的值由Furstenberg定理（因此积极的）的所有参数，但下（liminf）李雅普诺夫指数是等于零的一个密集的G（δ）零Hausdorff维数的参数集。作为我们的方法的副产品，我们提供了一个纯粹的几何证明的谱安德森本地化的离散薛定谔算子的随机势（包括Anderson-Bernoulli模型）的一维格子。(c)2020爱思唯尔公司All rights reserved.
We consider random products of SL(2,R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense G(delta) set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrodinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice. (c) 2020 Elsevier Inc. All rights reserved.