It has been conjectured that if a
locally compact group G has a continuous automorphism which is ergodic with
respect to Haar measure then G must be compact. This is true when G is
commutative or connected. In this paper further results in support of this conjecture
are presented. In particular, it is shown that the problem can be reduced to the
consideration of compactly generated, totally disconnected, locally compact groups
without compact, open, normal subgroups and that the conjecture holds for
many automorphisms of a certain class of such groups. Finally, the structure
of locally compact groups which admit ergodic affine transformations is
investigated.
