广义Baker变换的极限定理
Limit Theorems for Generalized Baker's Transformations
摘要：
本文研究了广义Baker变换的相关衰减和极限定理。我们的例子是单位正方形上的分段非一致双曲映射，它包含两条空间分离的无差异不动点线。我们得到了单位正方形上Lipschitz函数的锐混合率和单位正方形上Hölder观测量的极限定理。我们的一些极限定理表现出收敛到Hölder观测量的非正态稳定分布。我们观察到，任何偏度参数在$[-1,1]$的允许范围内的稳定分布，可以得到作为一个限制，并推导出一个明确的偏度参数之间的关系和沿线的中立不动点的Hölder观测值。本文首次将各向异性巴拿赫空间方法和算子更新理论应用于广义baker变换。我们的衰减相关的结果恢复玻色和默里的结果。我们的极限定理的结果是新的广义面包师变换。
英文摘要：
In this paper we study decay of correlations and limit theorems for generalized baker's transformations. Our examples are piecewise non-uniformly hyperbolic maps on the unit square that posses two spatially separated lines of indifferent fixed points. We obtain sharp rates of mixing for Lipschitz functions on the unit square and limit theorems for Hölder observables on the unit square. Some of our limit theorems exhibit convergence to non-normal stable distributions for Hölder observables. We observe that stable distributions with any skewness parameter in the allowable range of $[-1,1]$ can be obtained as a limit and derive an explicit relationship between the skewness parameter and the values of the Hölder observable along the lines of indifferent fixed points. This paper is the first application of anisotropic Banach space methods and operator renewal theory to generalized baker's transformations. Our decay of correlations results recover the results of Bose and Murray. Our results on limit theorems are new for generalized baker's transformations.
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