Nova S~rie BOLETIM DA SOCIEDADEBRASILEIRA DE MATEMATICA Bol. Soc. Bras. Mat., Vol. 28, N. 2, 197-210 (~) 1997, Sociedade Brasildra de Matemdtica Stable ergodicity in homogeneous spaces Jonathan Brezin and Michael Shub --Dedicated to the memory of Ricardo Mated Abstract. In this paper we prove that in the context of homogeneous spaces G/t? which satisfy a certain admissibility requirement, stable ergodicity of an affine diffeo- morphism implies that there is some hyperbolicity. Indeed, HB = G where H is the hyperbolically generated subgroup of G. 0. Introduction Our goal in this paper is to classify stably ergodic translations and affine maps on homogeneous spaces. We will assume that our spaces are of the form G/B where G is a connected Lie group and B is a closed subgroup which, in addition, is admissible in a certain technical sense (see below). For g C G let Lg denote left translation by g i.e. Lg(h) = 9h for all h E G. Then Lg induces a map on G/B which we call Lg as well. Given an automorphism A of G and g E G we call LgA : G --* G an attine diffeomorphism of G, we also denote this map by 9A. If A(B) = B then we continue to denote the induced map on G/B by LgA or gA and call it an affine diffeomorphism of G/B. For our discussion of ergodicity, we will assume that the Haar measure on G induces a finite measure on G/B which is invariant under left translation and that A : G/B ~ G/B is measure preserving. A measure preserving diffeomorphism is called ergodic if the only measurable invariant sets have measure zero or one. We say that an affine diffeomorphism aA of G/B is stably ergodic under perturbations by left translations if there is a neighborhood U of a in G such that a'A Received 23 Ocrober 1995.