DENSE ORBITS OF AFFINE AUTOMORPHISMS AND COMPACTNESS OF GROUPS S. G. DANI An ajfine automorphism T of a topological group G is a composite map T — l g o A where A is a continuous group automorphism of G and l g is the left translation of G by g e G; that is to say, l g (h) = gh for all heG. The object of this article is to prove the following result. THEOREM. Let G be a connected locally compact topological group. Suppose that there exist an ajfine automorphism T ofG and g o e G such that the orbit {T j (g 0 ) | j e Z} of T is dense in G. Then G is compact. This result on orbits of affine automorphisms has the following interesting corollary in ergodic theory. COROLLARY. Let G be a connected locally compact non-compact topological group. Let T be an ajfine automorphism ofG. Let \ibe a o-finite measure on G which is quasi-invariant and ergodic under the action of T. Then there exists a proper closed subset C of G such that n(G — C) = 0. In particular, the Haar measure of G is not ergodic with respect to any affine automorphism ofG. (We recall that, in the above notation, fi is said to be quasi-invariant if, for a Borel set E, ^(T'^E)) = 0 if and only if /i(£) = 0. Further it is said to be ergodic if, for a Borel set E, fi(T~ 1 (E) A E) = 0 implies that either fi(E) or /J.(G — E) is zero.) The question as to whether the Haar measure of a non-compact (locally compact) group G could be ergodic under a group automorphism was raised by P. R. Halmos (cf. [3; p. 29]). For the subclass of connected groups it was answered in the negative by R. Kaufman and M. Rajagopalan [4]. A proof of a slightly stronger result, but for affine automorphisms instead of automorphisms, appeared in [5]. We note however, that the proof offered in [5] for the general result, Theorem 3.1, is incorrect (cf. [2]). In [2] the present author studied the quasi-invariant measures of group automorphisms, and the results (cf. Theorem 3.7 and Corollary 3.8) imply the following. Let G be a connected locally compact non-compact group and A be a continuous group automorphism of G. Let fx be a measure on G which is quasi- invariant and ergodic under A. Then there exists a closed normal /4-invariant subgroup H of G and a compact subset C of G such that G/H is a non-compact Lie group and fi(G-CH) = 0. Further the action on CH/H induced by A extends to an action of a compact group. While for group automorphisms this result is stronger than the above-mentioned corollary, the proof is more complicated and its extension to affine automorphisms poses certain technical difficulties. The present proof is direct and independent of [2], though the method is similar. Received 30 March, 1981. [J. LONDON MATH. SOC. (2), 25 (1982), 241-245]