Dimensions of equilibrium measures on a class of planar self-affine sets

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2017-06-21 DOI:10.4171/JFG/85

J. Fraser, T. Jordan, Natalia Jurga

引用次数: 5

Abstract

We study equilibrium measures (Kaenmaki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure. In particular, we do this by showing that the Kaenmaki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.

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一类平面自仿射集上平衡测度的维数

我们研究了自仿射集上支持的平衡测度（Kaenmaki测度），该自仿射集是由作用在平面上的对角矩阵和反对角矩阵的有限集合生成的，并且满足强分离性质。我们的主要结果是，这些测度是精确的维数，并且维数满足Ledrapier-Young公式，该公式根据熵和Lyapunov指数以及测度的重要坐标投影的维数给出了维数的显式表达式。特别地，我们通过证明Kaenmaki测度等于有限类型的相关子移位上的两个Gibbs测度的（推进）之和来实现这一点。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.