Positive Lyapunov exponent for generic one-parameter families of unimodal maps

Let f _{a a∈A} be a C² one-parameter family of non-flat unimodal maps of an interval into itself and a* a parameter value such that (a) f_a* satisfies the Misiurewicz Condition, (b) f_a* satisfies a backward Collet-Eckmann-like condition, (c) the partial derivatives with respect to x and a of f _a ⁿ (x), respectively at the critical value and at a*, are comparable for large n. Then a* is a Lebesgue density point of the set of parameter values a such that the Lyapunov exponent of f_a at the critical value is positive, and f_a admits an invariant probability measure absolutely continuous with respect to the Lebesgue measure. We also show that given f_a* satisfying (a) and (b), condition (c) is satisfied for an open dense set of one-parameter families passing through f_a*.