Résume | I don't know the answer to this question, but I find it interesting and have some things to say about it. I'll discuss a one-parameter family within which pseudo-Anosovs (and generalized pseudo-Anosovs) form a countable dense subset, and describe the structure of the remaining maps in the family: they are called measurable pseudo-Anosovs (mpA). I'll also describe a way of taking quotients of surface diffeomorphisms - the 0-entropy equivalence - which, conjecturally, yields a mpA quotient. If the conjecture holds, it would follow from a result of Bonatti-Crovisier that mpAs form a C^1-residual subset of area-preserving surface diffeomorphisms. |