Finite and Infinite Rotation Sequences and Beyond

Abstract

The encoding of orbits attained from rigid rotations are investigated from different perspectives. In the first part of the thesis regularity conditions for irrational rotations will be studied in terms of their continued fraction expansions and a categorisation is achieved for continued fraction expansions which do not grow too fast. The second part focuses on the spectral properties of beta-transformations for beta sqrt(2). Here an explicit representation for the Bochner transform of autocorrelations stemming from Dirac combs derived from beta-transformations is achieved, which consists of a Lebesgue absolutely continuous part and a discrete part. The last part focuses on vague limits of these autocorrelations where beta tends to 1. Here a link to subshifts derived from rigid rotations will be established. The Bochner transform of these vague limits can be given explicitly in some cases and is shown to be either discrete, non-discrete singular to Lebesgue, or a mixture of both.