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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
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.