多重分形分析的热力学方法
The thermodynamic approach to multifractal analysis
摘要：
在多重分形分析中，大多数结果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饱和方法获得的。Barreira和Saussol引入了一个包含最重要的多重分形谱的一般框架，他们使用热力学方法建立了一致双曲背景下的多重分形形式，统一了许多现有的结果。我们扩展这个框架，适用于广泛的一类非一致双曲系统，包括相变的例子。在这个过程中，我们比较了这种热力学方法与饱和的方法，并给出了许多文献中的多重分形结果的调查。
英文摘要：
Most results in multifractal analysis are obtained using either a thermodynamic approach based on existence and uniqueness of equilibrium states or a saturation 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. In the process, we compare this thermodynamic approach with the saturation approach and give a survey of many of the multifractal results in the literature.
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