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.