Random beta -transformations

随机 beta 变换

• 批准号: 2902111
• 金额: --
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2902111
• 关键词: Random; beta; transformations

The doctoral educational programme consists of two interlinked research projects relating to the highly influential mathematical fields of fractal geometry, numeration systems and random dynamical systems. Project 1 will look at restricted digit sets for redundant number systems and perform a careful analysis of related affine fractals. Through utilising these affine fractals as quintessential examples of affine fractals, we will look at tackling the open question of when various notions of fractal dimensions coincide for affine fractals. Project 2 will examine means of defining natural extensions of random beta-expansions. Addressing questions such as if the resulting fundamental domain has positive Lebesgue measure, and if it gives rise to a tilling of Euclidean space: shedding new light on the still open Pisot conjecture.

博士教育课程由两个相互关联的研究项目组成，这些项目与极具影响力的数学领域--分形几何、计数系统和随机动力系统--有关。项目1将研究冗余数字系统的受限数字集，并对相关的仿射分数进行仔细的分析。通过使用这些仿射分形图作为仿射分形图的典型例子，我们将着眼于解决这样一个悬而未决的问题：对于仿射分形图，不同的分维概念何时重合。项目2将研究定义随机Beta扩展的自然扩展的方法。解决这样的问题，如所得到的基本域是否有正的勒贝格测度，以及它是否引起欧几里得空间的平铺：为仍然开放的皮索猜想提供新的曙光。