In [PS] it is conjectured that among the volume preserving C² diffeomorphisms of a closed manifold which have some hyperbolicity, the ergodic ones contain an open and dense set. In this paper we prove an analogous statement for skew products of Anosov diffeomorphisms of tori and circle rotations. Thus this paper may be seen as an example of the phenomenon conjectured in [PS]. The corresponding theorem for skew products of Anosov diffeomorphisms and translations of arbitrary compact groups is an interesting open problem.
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Adler et al. (1996) studied this question.