Thermodynamic Formalism and Dimension of Overlapping Fractal Measures

本课题将研究具有重叠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的降维。这可能是分形几何中最大的开放问题，也是最近取得重大进展的问题（最著名的是Hochman、Shmerkin、Varju和Breuillard）。最近的大部分进展都是通过遍历理论取得的。这个项目将继续在分形几何和遍历理论的界面上工作。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。最初的目标是将这个问题与热力学形式论中涉及环面上的加倍映射的问题联系起来。该项目的新颖之处在于在分形几何和热力学形式主义之间建立了新的联系，并解决了热力学形式主义中的新问题。

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.