Hyperbolic Attractors Which are Anosov Tori

Abstract

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 \(M^{n}\).