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

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We show that the time-1 map of an Anosov flow, whose strong-unstable foliation is C² smooth and minimal, is C² close to a diffeomorphism having positive central Lyapunov exponent Lebesgue almost everywhere and a unique physical measure with full basin, which is Cʳ stably ergodic. Our method is perturbative and does not rely on preservation of a smooth measure.