On existence of expanding attractors with different dimensions

Abstract

We prove that $n$-sphere ${\mathbb{S}}^n$, $n\ge 2$, admits structurally stable diffeomorphisms ${\mathbb{S}}^n\to {\mathbb{S}}^n$ with non-orientable expanding attractors of any topological dimension $d\in \{1,\dots ,[\frac{n}{2}]\}$ where $[x]$ is an integer part of $x$. One proves that $n$-torus ${\mathbb{T}}^n$, $n\ge 2$, admits structurally stable diffeomorphisms ${\mathbb{T}}^n\to {\mathbb{T}}^n$ with orientable expanding attractors of any topological dimension $1\le q\le n-1$. We also prove that given any closed $n$-manifold $M^n$, $n\ge 2$, and any $d\in \{1,\dots ,[\frac{n}{2}]\}$, 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.