Induced maps of hyperbolic Bernoulli systems

Let $(f,T^n,\mu)$ be a linear hyperbolic automorphism of the $n$-torus. We show that if $A\subset T^n$ has a boundary which is a finite union of $C^1$ 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 $C^1$ 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.