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We prove that an n -sphere S^n, n 2, admits structurally stable diffeomorphisms S^n^n with nonorientable expanding attractors of any topological dimension d\1, , n{2\} where x is the integer part of x. In addition, any n -sphere S^n, n 3, admits axiom A diffeomorphisms S^n^n with orientable expanding attractors of any topological dimension d\1, , n{3\}. We prove that an n -torus T^n, n 2, admits structurally stable diffeomorphisms T^n^n with orientable expanding attractors of any topological dimension d\1, , n-1\. We also prove that, given any closed n -manifold M^n, n 2, and any d\1, , n{2\}, there is an axiom A diffeomorphism f: M^n M^n with a d -dimensional nonorientable expanding attractor. Similar statements hold for axiom A flows.
Медведев et al. (Fri,) studied this question.