  We show that the time-1 map of an Anosov flow, whose strong-unstable
foliation is $C^2$ smooth and minimal, is $C^2$ close to a diffeomorphism
having positive central Lyapunov exponent Lebesgue almost everywhere and a
unique physical measure with full basin, which is $C^r$ stably ergodic. Our
method is perturbative and does not rely on preservation of a smooth measure.
