Hyperbolic absolutely continuous invariant measures for C r one-dimensional maps

For r > 1 , we show, using the Ledrappier–Young entropy characterization of SRB measures for non-invertible maps, that if a C r map f of the interval or the circle has its Lyapunov exponent greater than 1 r log ∥ f ′ ∥ ∞ on a set A of positive Lebesgue measure, then it admits hyperbolic ergodic invariant measures that are absolutely continuous with respect to the Lebesgue measure. We also show that the basins of these measures cover A Lebesgue-almost everywhere.