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

On the abundance of non-zero central Lyapunov exponents, physical measures and stable ergodicity for partially hyperbolic dynamics