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