Regularity of Aperiodic Subshifts and Connections to Intermediate beta-Transformations

Abstract

At the turn of this century Durand, Lagarias and Pleasants established that key features of minimal subshifts to be studied are linearly repetitive, repulsive and power free. In this thesis, we introduce generalisations and extensions of these features and establish a basic theory. Further, we study these new notions in the context of Sturmian subshifts and a family of aperiodic minimal subshifts stemming from Grigorchuk's infinite 2-group. In the second part, we study sequences of intermediate beta-transformations. Especially, we answer the question of how a sequence of corresponding normalised Parry measures converges as beta goes to one and we connect this convergence to Sturmian subshifts and the famous Thue-Morse sequence.