Random beta -transformations

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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.