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Abstract. We show that if a closed n-manifold M n (n ≥ 3) admits a structurally stable diffeomorphism f with an orientable expanding attractor Ω of codimension one, then M n is homotopy equivalent to the n-torus T n and is homeomorphic to T n for n = 4. Moreover, there are no nontrivial basic sets of f different from Ω. This allows us to classify, up to conjugacy, structurally stable diffeomorphisms having codimension one orientable expanding attractors and contracting repellers on T n , n ≥ 3.

We show that if f : M 3 → M 3 is an A-diffeomorphism with a surface two-dimensional attractor or repeller B and M 2 B is a supporting surface for B, then B = M 2 B and there is k ≥ 1 such that:i is homeomorphic to the 2-torus T 2 . 2) the restriction of f k to M 2 i (i ∈ {1, . . . , k}) is conjugate to Anosov automorphism of T 2 .