On structurally stable diffeomorphisms with codimension one expanding attractors

Abstract

We show that if a closed$n$-manifold$M^n$$(n\ge 3)$admits a structurally stable diffeomorphism$f$with an orientable expanding attractor$\Omega$of codimension one, then$M^n$is homotopy equivalent to the$n$-torus$T^n$and is homeomorphic to$T^n$for$n\ne 4$. Moreover, there are no nontrivial basic sets of$f$different from$\Omega$. This allows us to classify, up to conjugacy, structurally stable diffeomorphisms having codimension one orientable expanding attractors and contracting repellers on$T^n$,$n\ge 3$.