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We consider a family $${\{T_{r}: [0, 1] \circlearrowleft \}_{r\in[0, 1]}}$$ of Markov interval maps interpolating between the tent map $${T_{0}}$$ and the Farey map $${T_{1}}$$ . Letting $${\mathcal{P}_{r}}$$ denote the Perron–Frobenius operator of $${T_{r}}$$ , we show, for $${\beta \in [0, 1]}$$ and $${\alpha \in (0, 1)}$$ , that the asymptotic behaviour of the iterates of $${\mathcal{P}_{r}}$$ applied to observables with a singularity at $${\beta}$$ of order $${\alpha}$$ is dependent on the structure of the $${\omega}$$ -limit set of $${\beta}$$ with respect to $${T_{r}}$$ . The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.

We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $\alpha$-Farey maps. We provide a family of observables which are Riemann integrable, locally constant and of bounded variation, and for which the iterates of the transfer operator, when applied to one of these observables, is not asymptotic to a constant times the wandering rate on the first element of the partition $\alpha$. Subsequently, sufficient conditions on observables are given under which this expected asymptotic holds. In particular, we obtain an extension theorem which establishes that, if the asymptotic behaviour of iterates of the transfer operator is known on the first element of the partition $\alpha$, then the same asymptotic holds on any compact set bounded away from the indifferent fixed point.