On the Existence of Expanding Attractors with Different Dimensions

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.