Let (f, Tⁿ, mu) be a linear hyperbolic automorphism of the n-torus. We show that if A ⊂ Tⁿ has a boundary which is a finite union of C¹ submanifolds which have no tangents in the stable (E^s) or unstable (E^u) direction then the induced map on A, (f_A,A, mu_A) is also Bernoulli. We show that Poincáre maps for uniformly transverse C¹ Poincáre sections in smooth Bernoulli Anosov flows preserving a volume measure are Bernoulli if they are also transverse to the strongly stable and strongly unstable foliation.