Random beta -transformations

博士教育计划包括两个相互关联的研究项目，涉及分形几何，计数系统和随机动力系统的高度影响力的数学领域。项目1将研究冗余数系统的受限数字集，并对相关的仿射分形进行仔细分析。通过利用这些仿射分形作为仿射分形的典型例子，我们将探讨如何解决仿射分形的各种分形维数概念何时重合的问题。项目2将研究定义随机β-扩展的自然扩展的方法。解决的问题，如如果所产生的基本域有积极的勒贝格措施，如果它引起了一个耕作的欧几里德空间：新的光洒在仍然开放的皮索猜想。

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.