We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do not assume the existence of a Markov partition but only require the first image of the fundamental partition to be finite, we use canonical Markov extensions to first prove pointwise dual-ergodicity, which, together with an identification of wandering rates, leads to distributional limit theorems. We show that $T$ satisfies Rohlin's formula and prove a variant of the Shannon–McMillan–Breiman theorem. Moreover, we give a stronger limit theorem for the transfer operator providing us with a large collection of uniform and Darling–Kac sets. This enables us to apply recent results from fluctuation theory.