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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.
Медведев et al. (Thu,) studied this question.