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

Abstract

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 $T_r$. 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.

References in corpus (1)

• On the asymptotics of the $\alpha$-Farey transfer operator

Cited by in corpus (1)

• Strong laws of large number for intermediately trimmed Birkhoff sums of observables with infinite mean