Ergodic and combinatorial methods in fractal geometry

分形几何包括研究自相似和自仿射集，从动力学和一般的任何集合（通常在欧几里德空间）所产生的集合，其中豪斯多夫/包装维数是一个有趣的属性。许多关于自相似集和自仿射集的最新进展使用了遍历理论的思想和方法;许多在组合学（离散几何）中解决和未解决的问题在分形几何中有有趣的对应。该研究的主要部分将集中在自相似集和一个版本的tube-null属性，旨在找到自相似集的有效覆盖与少量的窄管，并调查某些biLipschitz不变量的后果。这项研究适合在数学分析研究领域，完全在数学科学的主题。

Fractal geometry includes the study of self-similar and self-affine sets, sets arising from dynamics and in general any sets (usually in Euclidean spaces) forwhich Hausdorff/packing dimension is an interesting property. Many recent advances about self-similar and self-affine sets use ideas and methods fromergodic theory; many solved an unsolved problems in combinatorics (discrete geometry) have interesting counterparts in fractal geometry. A main part of theproposed research will focus on self-similar sets and a version of the tube-null property, aiming at finding efficient coverings of self-similar sets with a smallnumber of narrow tubes and investigating the consequences for certain biLipschitz invariants. This research fits in the Mathematical Analysis research area, wholly within the Mathematical Sciences theme.