On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

We consider a family {T_r : [0, 1] → [0, 1] } _{r ∈ [0, 1]} of Markov interval maps interpolating between the tent map T₀ and the Farey map T₁.  Letting P_r denote the Perron–Frobenius operator of T_r, we show, for β ∈ [0,1] and α ∈ (0, 1) , that the asymptotic behaviour of the iterates of P_r applied to observables with a singularity at β of order α is dependent on the structure of the ω-limit set of β with respect to T_r. The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.