Doubly intermittent maps with critical points, unbounded derivatives and regularly varying tail

We consider a class of interval maps with up to two indifferent fixed points, an 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. Moreover, in the finite measure case, we give a formula for the measure-theoretical entropy and upper bounds for a very slow decay of correlations. This extends former work by Coates, Luzzatto, and Muhammad to maps with regularly varying tails. Particularly, we investigate the boundary case where the behaviour of the slowly varying function decides on the finiteness of the measure and on the decay of correlations.