Discrete real time flows with quasi-discrete spectra and algebras generated by expq(t)

Abstract

These notes (essentially unedited) were sent to W. Parry in 1964. The first two parts are complete and in a letter to Parry at that time Hahn indicated his intention to publish them. Evidently he did not manage to do this. The remainder of these notes represents an attempt to establish a theory of quasidiscrete spectra for discrete one-parameter flows. Hahn indicates the gaps and in a following note Parry clarifies his theory. The first part of these notes presents a characteristic example of a discrete one-parameter flow with quasidiscrete spectrum. Ergodicity, minimality and distality are established. The second part examines the Banach algebra of functions onR generated by {expq(t): q a real polynomial of degree <n+1} and shows that the shift isometries arise from a discrete one-parameter flow on its maximal ideal space Λ _n and that ifn is finite this flow is isomorphic to the example examined in the first part.