Title:

Ergodic theory of Gspaces

The thesis is in the form of three papers. In Paper I, affine transformations of a locally compact group are considered. In Part I, it is assumed that the group is not compact: it is shown that an affine transformation of an abelian or connected group cannot be ergodic unless the transformation is of one exceptional type. An attempt is made to obtain stronger conditions than nonergodicity. Part II deals with compact groups: it is shown that an affine transformation of a compact group is ergodic if and only if it has a dense orbit. For a connected group, alternative conditions are given. In particular, it is shown that an affine transformation of a Lie group cannot be ergodic unless the group is a torus. Papers II and III are concerned with the entropy theory of a measurepreserving transformation. The entropy of a transformation T (denoted by h(T)) was introduced by Kolmogorov in 1958 (and later modified by Sinai) as a 'nonspectral invariant': two transformations cannot be isomorphic unless they have the same entropy. Zero entropy has a special significance. In general, every transformation has a unique part with zero entropy; if this part is trivial, the transformation is said to have completely positive entropy. It is very useful to know that a transformation has completely positive entropy: such transformations are mixing of all orders and invertible transformations are Kolmogorov automorphisms. Paper II considers the question of completely positive entropy when the measure space of the transformation T is a Gspace for a compact separable group G. T is required to commute with Gaction: T.g = g.T for all g in G, where o is a group endomorphism of G onto G. TS(G) denotes the induced transformation on the space of Gorbits. It is proved that if T is weakly mixing (has a continuous spectrum) and TS(G) has completely positive entropy, then T has completely positive entropy. This theorem 'lifts' the property of completely positive entropy from the orbit space to the fundamental space. In Paper III, it is shown that under suitable conditions (which are not in fact very restrictive) h(T) = h(TS(G)) + h(o).
