Thermodynamic Formalism and Dimension of Overlapping Fractal Measures

热力学形式主义和重叠分形测度的维数

• 批准号: 2905612
• 金额: --
• 依托单位: University of Manchester
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2905612
• 关键词: Thermodynamic; Formalism; Dimension; Overlapping; Fractal

This project will study the dimension theory of self-similar measures with overlaps. In particular, we will study dimension drop in simple classes of overlapping self-similar measures. This is perhaps the biggest open problem in fractal geometry, and is one in which there has been substantial recent progress (most notably by Hochman, Shmerkin, Varju and Breuillard). Much of this recent progress has been made through ergodic theory. This project will continue to work at the interface of fractal geometry and ergodic theory.The student will focus on the case of self similar sets with three or more contractions, in which the contraction rate is 1/2 (projections of the Sierpinski gasket) or a reciprocal of a Pisot number. The initial goal is to link the question to problems in thermodynamic formalism involving the doubling map on the torus. The novelty of the project will lie both in making new connections between fractal geometry and thermodynamic formalism, and in solving new problems in thermodynamic formalism.

本计画将研究具有重迭的自相似测度的维数理论。特别地，我们将研究重叠自相似测度的简单类中的降维。这也许是分形几何中最大的开放性问题，并且最近取得了重大进展（最著名的是Hochman，Shmerkin，Varju和Breuillard）。最近的进展大部分是通过遍历理论取得的。本项目将继续研究分形几何和遍历理论的界面。学生将专注于具有三个或更多收缩的自相似集的情况，其中收缩率为1/2（Sierpinski垫片的投影）或Pisot数的倒数。最初的目标是链接的问题，在热力学形式主义的问题，涉及加倍映射的环面。该项目的新奇将在于使分形几何和热力学形式主义之间建立新的联系，并解决热力学形式主义中的新问题。