Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures

Source abstract

We consider the C 2 C^{2} set of C 2 C^{2} diffeomorphisms of the 2-torus T 2 \mathbb {T}^{2} , provided the conditions that the tangent bundle splits into the directed sum T T 2 = E s ⊕ E u T\mathbb {T}^{2}=E^{s}\oplus E^{u} of D f Df -invariant subbundles E s E^{s} , E u E^{u} and there is 0 > λ > 1 0>\lambda >1 such that ‖ D f | E s ‖ > λ \Vert Df|_{E^{s}}\Vert >\lambda and ‖ D f | E u ‖ ≥ 1 \Vert Df|_{E^{u}}\Vert \ge 1 . Then we prove that the set is the union of Anosov diffeomorphisms and diffeomorphisms approximated by Anosov, and moreover every diffeomorphism approximated by Anosov in the C 2 C^{2} set has no SBR measures. This is related to a result of Hu-Young.