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

Abstract

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 ' || $\infty$ on a set E 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 E Lebesgue-almost everywhere.