Using soft techniques we prove lower bounds for the maximum entropy of a system in a family in terms of the entropy of a random product of the systems in the family. We accomplish this for two families of immersions of the circle. For one family this is joint work with Leonel Robert and Enrique Pujals and the other Rafael de la Llave and Carles Simo. Time permitting we recall a two dimensional analog whose entropy properties are still unknown, but for which partial

results have been achieved in joint work with Francois Ledrappier, Carles Simo and Amie Wilkinson.