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.