Large derivatives, backward contraction and invariant densities

We will discuss the dynamics of smooth maps of the interval with finitely many non-flat critical points. We will show that if for such a map the derivatives of the iterates are sufficiently large at each critical value (depending on the number of critical points and their orders), then this map posses an abolutely continuous invariant measure. This is a joint work with H. Bruin, W. Shen and S. van Strien.