On existence of expanding attractors with different dimensions

We prove that n-sphere Sⁿ, n≥ 2, admits structurally stable diffeomorphisms Sⁿⁿ with non-orientable expanding attractors of any topological dimension d∈\1,…,[n/2]\ where $[x]$ is an integer part of x. One proves that n-torus Tⁿ, n≥ 2, admits structurally stable diffeomorphisms Tⁿⁿ with orientable expanding attractors of any topological dimension 1≤ q≤ n-1. We also prove that given any closed n-manifold Mⁿ, n≥ 2, and any d∈\1,…,[n/2]\, there is an axiom A diffeomorphism f: Mⁿ→ Mⁿ with a d-dimensional non-orientable expanding attractor. Similar statements hold for axiom A flows.