Doubly Intermittent Maps with Critical Points, Unbounded Derivatives and Regularly Varying Tail

We consider a class of doubly intermittent maps with critical points, unbounded derivative and regularly varying tails. Under some mild assumptions we prove the existence of a unique mixing absolutely continuous invariant measure and give conditions under which the measure is finite. This extends former work by Coates, Luzzatto and Mubarak to maps with regularly varying tails. Particularly, we look at the boundary case where the behaviour of the slowly varying function decides if the invariant measure is finite or infinite.