The multifractal analysis of Birkhoff averages and large deviations
The multifractal analysis of Birkhoff averages and large deviations
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DOI：
10.1887/0750308036/b1058c18
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发表时间：
2001-06
影响因子：
4
通讯作者：
Y. Pesin;H. Weiss
中科院分区：
文献类型：
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作者：
Y. Pesin;H. Weiss
对于nite型的单侧子移位，我们描述了H的Birkho遍历定理中例外集的ne结构旧的连续函数。我们显示了一个密切的联系与大偏差理论的Birkho平均数，我们提供了几个应用概率和数论，包括推广的问题Billingsly。我们研究的相空间的Birkho平均水平集的分解。我们发现，通常有不可数的许多密集的水平集，每个水平集进行辅助平衡措施，具有恒定的逐点维数（一种自相似性）。这些均衡测度，每一个都支持一个测度零点集，是我们分析的关键。Floris Takens是动力系统维数理论的先驱，也是动力学特征多重分形分析的领导者。我们把这本关于伯克霍平均数的多重分形分析的手稿献给他。让：+A！+ A是nite型拓扑混合单侧子移位，2C（+A;R）是连续函数.用（x）表示沿着点x的轨道的Birkho平均值，即，(x)= lim n！1 1 n n 1 X
For one-sided subshifts of nite type, we describe the ne structure of the exceptional set in the Birkho ergodic theorem for Holder continuous functions. We show an intimate connection with large deviation theory for Birkho averages, and we provide several applications to probability and number theory, including a problem popularized by Billingsly. We study the decomposition of the phase space into level sets of the Birkho average. We show that there are typically uncountably many dense level sets and that each level set carries an auxiliary equilibrium measure, with constant pointwise dimension (a type of selfsimilarity). These equilibrium measures, each supported on a measure zero set, are key to our analysis. Floris Takens has been a pioneer in the dimension theory of dynamical systems and a leader in the multifractal analysis of dynamical characteristics. We dedicate this manuscipt on the multifractal analysis of Birkho averages to him. Let : +A ! + A be an topologically mixing one-sided subshift of nite type [Wal], and 2 C( +A;R) a continuous function. Denote by (x) the Birkho average of along the orbit of the point x, i.e., (x) = lim n!1 1 n n 1 X