Group actions and ergodic theory on Banach function spaces

Abstract

This thesis is an account of our study of two branches of dynamical systems theory, namely the mean and pointwise ergodic theory. In our work on mean ergodic theorems, we investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces which includes Fr echet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems. Next we turn to the topic of pointwise ergodic theorems. We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of -compact locally compact Hausdor groups acting measure-preservingly on - nite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on di erent Banach function spaces, and how the properties of these function spaces in-

uence the weak type inequalities that can be obtained. Finally, we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages. Our investigations of these two parts of ergodic theory are uni ed by the techniques used - locally convex vector spaces, harmonic analysis, measure theory - and by the strong interaction of the nal results, which are obtained in greater generality than hitherto achieved.