The thermodynamic approach to multifractal analysis
The thermodynamic approach to multifractal analysis
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DOI：
10.1017/etds.2014.12
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发表时间：
2014-10-01
影响因子：
0.9
通讯作者：
Climenhaga, Vaughn
中科院分区：
文献类型：
--
作者：
Climenhaga, Vaughn
多重分形分析中的大多数结果都是使用基于平衡态的存在性和唯一性的热力学方法或基于某些版本的规范属性的轨道粘合方法获得的。 Barreira 和 Saussol 引入了包含最重要的多重分形谱的通用框架，他们使用热力学方法在均匀双曲设置中建立了多重分形形式主义，统一了许多现有结果。我们扩展了这个框架，以应用于广泛的非均匀双曲系统，包括相变的例子，并为已经使用轨道粘合方法研究的许多例子获得了新的结果。我们比较了热力学和轨道粘合方法，并对文献中的许多多重分形结果进行了调查。
Most results in multifractal analysis are obtained using either a thermodynamic approach based on the existence and uniqueness of equilibrium states or an orbit-gluing approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions, and obtain new results for a number of examples that have already been studied using the orbit-gluing approach. We compare the thermodynamic and orbit-gluing approaches and give a survey of many of the multifractal results in the literature.