On ergodic ZZ-actions on Lie groups by automorphisms

In response to a question raised by Halmos in his book on ergodic theory ([10], page 29) it was proved that a locally compact group admits a (bicontinuous) group automorphism acting ergodically (with respect to the Haar measure as a quasiinvariant measure) only if it is compact (see [9] for historical details and a generalisation to affine transformations; see [5] for the case of Lie groups). A substantial part of ergodic theory is now being extended to actions of ZZ (the group of integer d-tuples) and it is natural in this context to ask which locally compact groups admit ergodic ZZ-actions by automorphisms. In this note we address the question for connected Lie groups, (and more generally for almost connected locally compact groups). Unlike in the case of d = 1, even for ZZ a connected Lie group admitting such an action need not be compact; e.g. the group IR of real numbers, the automorphism group consists of multiplications by non-zero real numbers and it has subgroups isomorphic to ZZ which are dense, and hence act ergodically on IR. The example readily generalises to actions of higher rank abelian groups, on higher dimensional vector spaces; more generally the connected abelian Lie group IR×TT (where TT denotes the m-dimensional torus) admits ergodic ZZ-actions by automorphisms for sufficiently large d, for any n ≥ 0 and m 6= 1 (see § 6 for precise results in this respect). We show here, in particular, that the general class of connected Lie groups with such actions is not much larger; we assume only existence of a dense orbit for the action, a condition which is satisfied if the action is ergodic; the condition however turns out to be equivalent to ergodicity in the present instance (see Theorem 1.1). However the abelian groups do not exhaust the class, and in fact there exist nonabelian Lie groups with ergodic ZZ-actions on them by automorphisms (see § 6.4).