Hyperbolic Attractors Which are Anosov Tori

We consider a topologically mixing hyperbolic attractor $\Lambda\subset M^n$ for a diffeomorphism $f:M^n\to M^n$ of a compact orientable $n$-manifold $M^n$, $n>3$. Such an attractor $\Lambda$ is called an Anosov torus provided the restriction $f|_{\Lambda}$ is conjugate to Anosov algebraic automorphism of $k$-dimensional torus $\mathbb T^k$. We prove that $\Lambda$ is an Anosov torus for two cases: 1) $\dim{\Lambda}=n-1$, $\dim{W^u_x}=1$, $x\in\Lambda$; 2) $\dim\,\Lambda=k,\,\dim\, W^u_x=k-1,\,x\in\Lambda$, and $\Lambda$ belongs to an $f$-invariant closed $k$-manifold, $2\leqslant k\leqslant n$, topologically embedded in $M^n$.