具有奇异性的拟周期余圈的大偏差
Large deviations for quasi-periodic cocycles with singularities
摘要：
我们推导了遍历多频环面平移上线性余循环的大偏差型（LDT）估计。这些模型被称为准周期上循环。我们对该模型作如下假设：平移向量满足一般丢番图条件，纤维作用量由多元矩阵值解析函数给出，且该函数不恒奇异.这里得到的LDT估计依赖于上循环上的一些统一测量。我们在[9]中得到的关于李雅普诺夫指数（LE）和Oseledets过滤和分解的连续性的一般结果是适用的，并且我们得到了在李雅普诺夫谱中存在间隙时这些量的局部弱Holder连续性。这项工作的主要新特点是允许依赖于几个变量的上循环具有奇点，即不可逆点。这就需要仔细分析的零点集的某些解析函数的几个变量和奇异性（即负无穷大值）的pluri-下调和函数有关的迭代的上循环。在一个变量的情况下，这种方法的改进导致更强的LDT估计，进而导致LE、Oseledets过滤和Oseledets分解的更强的、接近Holder连续性模量。这是我们即将出版的研究专著[9]中一章的草稿。
英文摘要：
We derive large deviations type (LDT) estimates for linear cocycles over an ergodic multifrequency torus translation. These models are called quasi-periodic cocycles. We make the following assumptions on the model: the translation vector satisfies a generic Diophantine condition, and the fiber action is given by a matrix valued analytic function of several variables which is not identically singular. The LDT estimates obtained here depend on some uniform measurements on the cocycle. Our general results derived in [9] regarding the continuity properties of the Lyapunov exponents (LE) and of the Oseledets filtration and decompositions are then applicable, and we obtain local weak-Holder continuity of these quantities in the presence of gaps in the Lyapunov spectrum. The main new feature of this work is allowing a cocycle depending on several variables to have singularities, i.e. points of non invertibility. This requires a careful analysis of the set of zeros of certain analytic functions of several variables and of the singularities (i.e. negative infinity values) of pluri-subharmonic functions related to the iterates of the cocycle. A refinement of this method in the one variable case leads to a stronger LDT estimate and in turn to a stronger, nearly-Holder modulus of continuity of the LE, Oseledets filtration and Oseledets decomposition. This is a draft of a chapter in our forthcoming research monograph [9].
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