On the convergence to equilibrium of unbounded observables under a   family of intermittent interval maps
  We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ 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 $T_{r}$. Having a singularity it seems that such observables do not fall into any of the function classes on which convergence to equilibrium has been previously shown. 
Johannes Kautzsch, Marc Kesseb\"ohmer, Tony Samuel
math.DS