  We prove that $n$-sphere $\mathbb{S}^n$, $n\geq 2$, admits structurally
stable diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with non-orientable
expanding attractors of any topological dimension
$d\in\{1,\ldots,[\frac{n}{2}]\}$ where $[x]$ is an integer part of $x$. One
proves that $n$-torus $\mathbb{T}^n$, $n\geq 2$, admits structurally stable
diffeomorphisms $\mathbb{T}^n\to\mathbb{T}^n$ with orientable expanding
attractors of any topological dimension $1\leq q\leq n-1$. We also prove that
given any closed $n$-manifold $M^n$, $n\geq 2$, 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 non-orientable expanding attractor. Similar
statements hold for axiom A flows.
